CASE STUDY
Bulle in o Enginee ing Geology and he En i onmen (2025) 84:471
h ps://doi.o g/10.1007/s10064-025-04513-7
In oduc ion
Rock all isk analysis is an essen ial aspec in he man-
agemen o moun ainous egions, whe e geohaza ds pose
signi ican h ea s o in as uc u e, human li e, and eco-
nomic ac i i ies (Hung e al. 1999). D i en by he scien-
i ic ad ancemen s (Mo gens e n 2018; Fell e al. 2005;
Co ominas e al. 2014; Maccio a e al. 2017), quan i a i e
isk assessmen (QRA) echniques ha e become inc eas-
ingly adop ed by Public Au ho i ies o e he pas decade
o decision-making in landslide-p one a eas (Ma chelli
e al. 2022a). The abili y o compa e isk le els agains
de ined h esholds enables mo e cos -e ec i e managemen
s a egies.
Resea ch on isk quan i ica ion encompasses all aspec s
o he ock all phenomenon, om he iden i ica ion o
uns able a eas and block de achmen o hei in e ac-
ion wi h u banised en i onmen s (Sca ia e al. 2020). As
wi h any complex phenomenon, he quali y o inpu da a
Maddalena Ma chelli
[email p o ec ed]
Vale io De Biagi
[email p o ec ed]
Ma co Paganone
[email p o ec ed]
Da ide Be olo
[email p o ec ed]
1 Depa men o En i onmen , Land and In as uc u e
Enginee ing, Poli ecnico di To ino, Co so Duca degli
Ab uzzi, 24, 10129 To ino, I aly
2 Depa men o S uc u al, Geo echnical and Building
Enginee ing, Poli ecnico di To ino, Co so Duca degli
Ab uzzi, 24, 10129 To ino, I aly
3 Geological O ice, Regione Au onoma Valle d’Aos a, Via
P omis, 2, 11100 Aos a, I aly
Abs ac
The quan i ica ion o ock all isk along an in as uc u e is a undamen al achie emen o an app op ia e managemen o
moun ain oads and i consis s o a ious s eps: om he iden i ica ion o he sou ces and sizes o he po en ial uns able
blocks, o he ajec o y analysis and, inally, he quan i ica ion o he e ec s on he elemen s a isk. The deg ee o knowl-
edge o he slope and he p e ious ock all e en s p o ides a solid base o he calcula ion and a p ecise isk quan i ica ion
can be ob ained o a eas o limi ed ex en . On he con a y, when mo ing o a eas o la ge ex en he p e ious s eps canno
be comple ely achie ed and he isk has o be compu ed by conside ing he e ec s o he limi ed knowledge. To his aim,
he pape de ails a p ocedu e o include he e ec o unce ain ies in o he quan i ica ion o he socie al isk along a oad.
The p oposed me hod is a hyb id quan i a i e app oach ha in eg a es elemen s o likelihood-based, uzzy, and Bayesian
me hodologies. I is speci ically designed o ock all isk assessmen s o e ex ensi e a eas unde condi ions o limi ed
da a a ailabili y. To add ess epis emic unce ain y, he me hod p ima ily in ol es assigning likelihoods o he equency
o blocks eaching he oad, based on his o ical da a, in o de o es ima e a ange o po en ial isks and hei associa ed
p obabili ies. Alea o y unce ain y inhe en in he phenomenon is handled using Mon e Ca lo p obabilis ic echniques. To
explain he a ious s eps in he analysis, he p oposed app oach is applied o a s udy case consis ing o a 7.5 km long
ou is ic oad subjec ed o ock all haza d in Aos a Valley, in he No hwes e n I alian Alps, conside ing di e en possible
a ic scena ios. I is shown ha he me hod is sui able o de e mine he isk when he knowledge o he a ea is limi ed.
Keywo ds Rock all · Quan i a i e analysis · Road · Haza d · Risk assessmen · Hyb id me hods · Unce ain ies
Recei ed: 8 Oc obe 2024 / Accep ed: 18 Sep embe 2025
© The Au ho (s) 2025
A hyb id app oach o quan i ying ock all isk wi h limi ed knowledge:
a case s udy in Aos a Valley
MaddalenaMa chelli1· Vale ioDe Biagi2· Ma coPaganone3· Da ideBe olo3
1 3
Bulle in o Enginee ing Geology and he En i onmen (2025) 84:471
signi ican ly in luences he eliabili y o ou pu s. Accu a e
p edic ions ega ding sou ce zones and he e olu ion o
ock all e en s equi e high- esolu ion opog aphic da a,
as well as de ailed geological, geomo phological, me eo o-
logical, and li hological in o ma ion, including ege a ion
co e and land use (Wei e al. 2014; Lan e al. 2010; Kanno
e al. 2023). Despi e p og ess in modelling and assessmen
echniques, unce ain y emains a pe asi e challenge in
p edic ing and mi iga ing ock all haza ds. These unce ain-
ies s em om a ious sou ces: he occu ence and size o
eleased blocks, hei dynamics along he slope (including
agmen a ion (Ma chelli e al. 2025)), and hei impac on
exposed elemen s.
Fo ins ance, he quan i ica ion o elease p obabili y
emains deba ed, as i is in luenced by nume ous poo ly
unde s ood a iables (D’Ama o e al. 2016; Moos e al.
2022). Consequen ly, obse a ional echniques, e.g. pe i-
odic su eys (Bude a e al. 2016; Ma asci e al. 2018; S un-
den e al. 2015) o con inuous moni o ing (Zimme and
Si a 2015; Feng e al. 2021), a e adop ed o es ima e e en
equency. Howe e , hese me hods ypically cap u e only
ecen ac i i y. His o ical in en o ies p o ide aluable long-
e m and la ge-scale eco ds o ock all ac i i y (Dussauge
e al. 2003; Hung e al. 1999), hough hey a e o en biased
due o unobse ed e en s (Volkwein e al. 2011). De Biagi
e al. (2017) add essed his issue by in oducing a h esh-
old olume in obse a ions. While ege a ion damage can
indica e pas e en s (T appmann e al. 2013; Fa illie e al.
2017), such e idence is gene ally limi ed o majo occu -
ences. Recen ly, Fa acque e al. (2021) in eg a ed dend o-
ch onological da a and deposi obse a ions om ock all
ba ie s wi h 3D modelling o es ima e he equency o di -
e en block olumes. Block olume is s ongly in luenced
by discon inui y se cha ac e is ics. Umili e al. (2023) p o-
posed a new o mula, based on Palms øm’s exp ession, o
accoun o alea o y a iabili y in eal ock aces. Volume
and shape a e key inpu s o p opaga ion analysis, which
is u he a ec ed by ock–soil in e ac ions (Aglia di and
C os a 2003; C os a and Aglia di 2004). Li and Lan (2015)
e iewed he unce ain ies in ajec o y modelling, high-
ligh ing hei dependence on model selec ion, assump ions,
and inpu accu acy.
Risk quan i ica ion, esul ing om elease, p opaga ion,
and in e ac ion, is inhe en ly a ec ed by hese unce ain ies.
Add essing unce ain ies is a challenge ha ex ends well
beyond he ock all sec o ; disciplines such as economics
and he social sciences ha e de eloped sui able app oaches
o inco po a e bo h alea o y and epis emic unce ain ies in o
isk assessmen (Sho idge e al. 2017).
In ock all amewo k, alea o y unce ain y, ha depends
on he inhe en andom a iabili y o p ocesses, is in lu-
enced by ac o s such as li hology, andom dis ibu ion o
ac u es in he ock (Wang e al. 2012, 2014a). The unce -
ain y in he size o he eleased block has gene ally been
ackled by conside ing di e en olumes and assigning o
each o hem hei e u n pe iod acco ding o a olume-
e u n pe iod law. Al hough he e a e e iden di icul ies
in es ima ing he ela ionship in he sou ce a ea (Wang
e al. 2014a), he olume- equency law can be ela ed o a
powe -law (Dussauge e al. 2003; Han z e al. 2016; Mac-
cio a e al. 2020). Such app oach has been adop ed by he
Au ho s in quan i ying he eliabili y o a ock all ba ie
(De Biagi e al. 2020; Ma chelli e al. 2020a, 2021). To
educe he impac o alea o y unce ain ies on he p opaga-
ion, say on he ajec o y analysis, a p obabilis ic app oach
is hus eques ed and a la ge numbe o h ows is neces-
sa y. In pa icula , a Mon e Ca lo app oach is widely used in
ajec o y analyses and haza d assessmen o add ess alea-
o y unce ain y. By epea edly sampling om p obabili y
dis ibu ions assigned o he inpu a iables, Mon e Ca lo
me hods allow o he es ima ion o he likelihood and ange
o possible ou comes, such as un-ou dis ances o impac
ene gies (Do en 2003; Maccio a e al. 2015). I is wo h
men ioning ha despi e widely adop ed o model alea o y
unce ain y, Mon e Ca lo app oach has no able limi a ions,
as i equi es high-quali y inpu da a, and poo -quali y da a
can lead o misleading esul s (Do en 2003). Mo eo e ,
many ock all ajec o y models simpli y physics (e.g., con-
s an es i u ion coe icien s), educing ealism when ampli-
ied h ough s ochas ic sampling (Chen e al. 2013).
Epis emic unce ain y in ock all modelling a ises om
limi ed knowledge abou key ac o s such as sou ce loca-
ion, e en equency, p opaga ion me hods, and pa ame e
selec ion. This unce ain y is compounded by he di icul y
in di ec ly measu ing essen ial pa ame e s and he imp ac i-
cali y o ull-scale expe imen s. As a esul , signi ican gaps
emain in unde s anding ma e ial beha iou , block geom-
e y, slope opog aphy, and block-slope in e ac ion mechan-
ics. Thus, bo h da a- ela ed and model- ela ed unce ain ies
all unde he ca ego y o epis emic unce ain y (Ranga a-
jhala e al. 2012). Fo ins ance, Mineo (2020) compa ed
a ious isk analysis echniques, highligh ing di e ences
in he le el o de ail, he ea men o unce ain y, and he
assump ions equi ed. To accoun o i , a ious sugges ions
a e possible, also in o he disciplines, including compu e
science and sa e y enginee ing. As an example, dealing wi h
QRA o a gene ic haza dous e en and a gene ic isk, A en
(2008) has p oposed o include semi-quan i a i e unce ain y
ac o s as an addi ional componen o he isk analysis wi h
he speci ic pu pose o s a e he knowledge ha suppo s he
esul . Lempe e al. (2006) ha e epea ed he same analy-
sis se e al imes wi h sligh ly di e en ini ial condi ions o
assess he ou pu s and check o he e ec s o unce ain y.
Besides adi ional models, uzzy logic app oaches can be
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Bulle in o Enginee ing Geology and he En i onmen (2025) 84:471
adop ed (De Ru and Elo 1996), p o iding a amewo k o
handling agueness and imp ecision in inpu da a. In uzzy
logic, membe ship unc ions a e used o exp ess he deg ee
o u h o con idence in quali a i e o unce ain in o ma-
ion, such as he likelihood o a slope being uns able o he
se e i y o po en ial impac s, a he han elying on bina y
o c isp alues. These unc ions allow o a mo e lexible
ep esen a ion o expe knowledge and subjec i e assess-
men s wi hin he isk analysis p ocess. Building on his
amewo k, uzzy numbe s, i.e. a speci ic ype o uzzy
se used o ep esen unce ain nume ical quan i ies, ha e
been applied o model epis emic unce ain ies in ock all
simula ions. Fo ins ance, Tu in e al. (2009) p oposed he
use o uzzy numbe s o ep esen unce ain ies in ma e ial
p ope ies and boulde geome y. This app oach enables he
quan i ica ion and p opaga ion o unce ain y h ough uzzy
ans o ma ion me hods, which can be used o de i e anges
o key pa ame e s such as coe icien s o es i u ion, essen-
ial o simula ing block-slope in e ac ion beha iou . Ne e -
heless, while uzzy logic models o e a lexible amewo k
o inco po a ing expe judgmen and handling imp ecise,
sca ce o quali a i e da a, hey also p esen no able limi a-
ions, including he subjec i i y in de ining membe ship
unc ions and ules, which can a ec ep oducibili y. Addi-
ionally, uzzy models lack a p obabilis ic ounda ion, mak-
ing i di icul o quan i y isk in e ms o likelihood o o
in eg a e hem wi h physically based models o dynamic
p ocesses like block ajec o ies (Shang and Kossen 2013).
A p obabilis ic amewo k ha allows o he explici inco -
po a ion and ep esen a ion o epis emic unce ain ies in
isk analysis elies in Bayesian ne wo ks, a model ha ep-
esen s a se o a iables and hei condi ional dependencies,
and he associa ed p ocess o Bayesian in e ence, which
upda es p obabili ies as new da a becomes a ailable (Fen-
on and Neil 2018; Kaikkonen e al. 2021). In he con ex
o ock all haza d assessmen , Bayesian in e ence p o ides
hus a sys ema ic way o upda e p io belie s on many inpu
pa ame e s subjec o epis emic unce ain y due o limi ed
da a o measu emen challenges as new e idence becomes
a ailable, esul ing in pos e io dis ibu ions ha e lec
imp o ed knowledge. By in eg a ing bo h expe judg-
men and obse a ional da a, Bayesian ne wo ks allow o
he dynamic e inemen o p obabili y dis ibu ions, esul -
ing in pos e io es ima es ha e lec imp o ed knowledge.
Howe e , despi e hei ad an ages, he p ac ical applica-
ion o Bayesian ne wo ks in geohaza d modelling p esen s
se e al challenges. These include he need o subs an ial
expe inpu o de ine he ne wo k s uc u e and assign p io
p obabili y dis ibu ions, which can be subjec i e and di -
icul o alida e in da a-sca ce en i onmen s (Zheng e al.
2021). Acqui ing su icien and eliable da a o suppo
in e ence is o en p oblema ic, especially in moun ainous o
inaccessible egions whe e ock all and landslide e en s a e
unde epo ed o poo ly documen ed (Hwang e al. 2024).
Mo eo e , he quali y o he esul s is highly dependen on
he accu acy o he condi ional p obabili y ables and he
assump ions embedded in he model. Incomple e o biased
da a can lead o misleading pos e io dis ibu ions, educ-
ing he obus ness o he isk assessmen . These limi a ions
highligh he impo ance o he in eg a ion o expe knowl-
edge wi h empi ical da a o ensu e meaning ul and ac ion-
able ou comes in geohaza d isk analysis (Ca denas e al.
2022).
To enhance p edic i e accu acy and manage epis emic
unce ain ies, machine lea ning (ML) echniques, anging
om decision ees o deep lea ning, a e inc easingly being
applied o ock all haza d and isk assessmen (Fa makis
e al. 2022, 2025; Fanos e al. 2020; Chanu e al. 2024).
Fo example, deep lea ning models can iden i y hidden pa -
e ns wi hin la ge da ase s, he eby compensa ing o gaps
in expe knowledge and educing epis emic unce ain y
(Abake e al. 2023). Howe e , alea o y unce ain y emains
mo e challenging o add ess, e en hough some s udies
a emp o inco po a e p obabilis ic amewo ks o ensemble
modelling o pa ially cap u e his a iabili y (Wang e al.
2014b). Despi e hei p omise, ML-based me hods ace
se e al challenges. Fi s , he si e-speci ic na u e o ock-
alls limi s he gene alizabili y o ained models. Second,
he sca ci y and imbalance o labelled da a (e.g., ela i ely
ew ock all e en s compa ed o non-e en s) can bias p e-
dic ions. Thi d, in e p e abili y emains a conce n–pa icu-
la ly o deep lea ning models, which a e o en pe cei ed
as “black boxes”. This necessi a es ca e ul calib a ion and
alida ion o ensu e model eliabili y and s akeholde us
(He e al. 2024).
In si ua ions whe e quan i a i e da a a e sca ce o incom-
ple e, o in la ge-scale assessmen s and ope a ional se ings
whe e de ailed simula ions may be imp ac ical due o ime
o da a cons ain s, likelihood-based sco ing sys ems a e
widely adop ed in ock all haza d assessmen . These me h-
ods a e o en quali a i e in na u e, elying on expe judg-
men and ield obse a ions o assign likelihood a ings o
po en ial ock all e en s h ough s uc u ed checklis s o
a ing ma ices (Fe a i e al. 2016; Fa me e al. 2023; Pan-
dey e al. 2025).
Finally, Wang e al. (2014a) ha e add essed bo h alea-
o y and epis emic unce ain ies by assigning coe icien s o
a ia ion o inpu a iables and applying a i s -o de sec-
ond-momen me hod o p opaga e a iance. They modelled
he elease p ocess using a powe -law ela ionship be ween
block size and equency, o e ing a s uc u ed p obabilis ic
amewo k. Howe e , his app oach equi es well-cha ac-
e ised inpu da a, which may limi i s applicabili y in da a-
sca ce en i onmen s. Ac oss he key s udies, Maccio a and
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Page 3 o 21 471
Bulle in o Enginee ing Geology and he En i onmen (2025) 84:471
leng h, o u he manage he isk along he in as uc u e.
The in o ma ion on he geological con ex was ex emely
limi ed: nei he speci ic su ey on he po en ial sou ce
zones no geomechanical analyses we e a ailable. Gi en
he ex ensi e size o he s udy a ea, la ge-scale emo e
sensing is nei he economically iable no p ac ically ea-
sible. As a esul , simpli ica ions and assump ions ega ding
po en ial block olumes and equencies i he e en s a e
necessa y, since acqui ing de ailed knowledge is no pos-
sible (Sca ia e al. 2020). Such le el o de ail, which can
be achie ed when he s udy a ea is limi ed in size, canno
be eached when mo ing o a la ge ex en , as in he p esen
s udy case. In his wo k, a speci ic isk assessmen me hod
based on E en T ee Analysis (ETA), de eloped by he
Au ho s (Ma chelli 2020; Ma chelli e al. 2022a), is applied.
This me hod inco po a es he p oposed hyb id app oach o
add ess unce ain ies, pa icula ly hose ela ed o he e-
quency o block impac s on elemen s a isk, which eme ges
as he mos ele an pa ame e in he o e all isk e alua ion.
The pape de ails he pe o med calcula ions, discusses he
unde lying hypo heses, and epo s he esul s o he isk
analysis.
The pape is o ganised as ollow. Fi s , he undamen als
o ock all isk assessmen on ehicula oads a e p esen ed
(Sec ion “Basics o isk calcula ion on oads”). The me hod-
ology o add essing unce ain ies, along wi h he unde ly-
ing assump ions, is in oduced in Sec ion “Me hodology”.
Sec ion “Case s udy” is en i ely dedica ed o he desc ip ion
o he case s udy and he p esen a ion o he analysis esul s.
The e ec s o unce ain y a e discussed, and sugges ions o
isk managemen a e p o ided in Sec ion “Discussions”.
Finally, he conclusions (Sec ion “Conclusions”) summa-
ize he main s eps o he analysis.
Basics o isk calcula ion on oads
The p esen sec ion summa izes he basics concep s o
quan i a i e isk assessmen wi h a pa icula ocus on an
app oach o quan i y ock all isk on oads, p e iously
de eloped by he Au ho s (Ma chelli 2020; Ma chelli e al.
2022a). This app oach cons i u es he analy ical amewo k
employed in he p esen s udy, whose a iables a e ea ed
using me hods ha accoun o bo h epis emic and alea o y
unce ain ies, inco po a ing likelihood-based easoning and
s a is ical analysis. Thus, he p esen ed o mulae will be
ecalled in he me hodology. Gene ally speaking, he isk
is de e mined as unc ion o he haza d, he exposu e, he
ulne abili y, and he alue (capaci y) (Co ominas e al.
2014; Fell e al. 2008), and i s calcula ion should accoun
o he p e iously men ioned la ge unce ain ies in he phe-
nomenon (Li and Lan 2015; Be en e al. 2018). Dealing
co-au ho s ha e p og essi ely e ined hei app oach o
managing unce ain y in slope isk assessmen . In he 2016
ailway case s udy (Maccio a e al. 2016b), alea o y unce -
ain y has been add essed h ough Mon e Ca lo simula ions,
while epis emic unce ain y has been managed ia expe
judgemen and bounding es ima es. The 2020 Canmo e
highway s udy (Maccio a e al. 2020) has adop ed a p oba-
bilis ic QRA amewo k, inco po a ing mul iple elease sce-
na ios (e.g., block size, equency), sensi i i y analysis, and
he use o uppe and lowe bounds o unce ain pa ame e s
o add ess bo h ypes o unce ain y. Ne e heless, bo h
s udies ely on de ailed si e-speci ic da a and expe calib a-
ion, making hem mo e sui able o local-scale applica ions
wi h su icien in o ma ion. In Maccio a (2023), he au ho s
shi ed he ocus owa ds he documen a ion o unce ain y
sou ces and he in luence o en i onmen al a iabili y, pa -
icula ly unde clima e change. While concep ually obus ,
his amewo k may be less e ec i e o de ailed design o
mi iga ion planning wi hou suppo ing da a.
In he p esen s udy, we p opose a hyb id quan i a i e
me hod ha in eg a es elemen s o likelihood-based, uzzy,
and Bayesian app oaches, speci ically designed o la ge-
scale ock all isk assessmen s unde condi ions o sca ce
da a. This app oach add esses a c i ical gap in cu en
me hodologies, which a e o en ei he quali a i e o equi e
ex ensi e da ase s, by enabling p obabilis ic easoning and
unce ain y quan i ica ion e en when empi ical da a a e lim-
i ed. Ou model assigns a likelihood, exp essed as a p ob-
abili y, o he mos c i ical sou ce o epis emic unce ain y
in ock all isk assessmen , i.e. he equency wi h which a
ock block impac s an elemen a isk. This p obabili y is
de i ed om his o ical e en da a bu can be dynamically
upda ed as new in o ma ion becomes a ailable, inco po-
a ing p io knowledge in a Bayesian sense. As such, he
me hod occupies a concep ual space be ween uzzy logic
and Bayesian in e ence: i e ains he in e p e abili y and
lexibili y o uzzy app oaches while enabling p obabilis-
ic upda ing ypical o Bayesian amewo ks. In pa allel,
alea o y unce ain ies a e add essed h ough a p obabilis-
ic app oach using Mon e Ca lo sampling echniques, wi h
pa icula e e ence o he p opaga ion phase o ock all
modelling, allowing o a obus es ima ion o impac p ob-
abili ies and spa ial haza d me ics. The hyb id s uc u e o
he me hod hus ensu es comp ehensi e ea men o bo h
epis emic and alea o y unce ain ies, making i pa icula ly
well-sui ed o la ge-scale applica ions whe e unce ain y is
high and da a a ailabili y is limi ed.
This s udy o igina es om a speci ic eques o he Local
Au ho i y o ha ing a gene al quan i ica ion o he socie al
isk, i.e. p obabili y o ha ing a ali ies, along a 7.5 km
ou is ic oad in No hwes e n I alian Alps, which expe i-
enced ock all p oblems in he pas along almos i s en i e
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471 Page 4 o 21
Bulle in o Enginee ing Geology and he En i onmen (2025) 84:471
p obabili ies o each scena io leading o a ic im among he
a elle s.
The en i e oad can be di ided in o po ions o ypi-
cally 50 o 100 me e s in leng h, which a e homogeneous in
e ms o a ic condi ions (e.g., speed, sigh dis ance, g adi-
en , e c.). Figu e 1 epo s he p oposed me hod, which is
applied o each k h po ion. Gi en he occu ence o he
ini ia ing e en , he scena ios e e s o a a al acciden as
ou come, and, hus, o each k h po ion o oad, he p ob-
abili y o ha e a a al acciden , de ined
PFk
is:
PFk=PFk
i+PFk
s+PFk
d,
(1)
whe e he subsc ip s i, s, and d e e o he case ha (i) a
mo ing ehicle is hi by he alling block, o (s) a mo ing
ehicle impac s on he block s opped on he oad, o (d) a
mo ing ehicle skids o damages on he oad caused by he
ebounding o he block on i . Fo simplici y o no a ion, he
equa ions ha ollow do no explici ly ecall he supe sc ip
k h indica ing he po ion o oad. The occu ence o he i s
scena io, i.e. he block hi s he ehicle, equi es he con em-
po anei y in ime and in space o he alling block and he
mo ing ehicle which is exp essed by he empo al-spa ial
p obabili y
P(S:T)i
:
P(S:T)i=P(T:V)iP(S:V)i,
(2)
being
P(T:V)i
and
P(S:V)i
he empo al and spa ial p oba-
bili ies, espec i ely. As de ailed in Ma chelli e al. (2022a),
hese p obabili ies can be compu ed as he p oduc o he
p obabili y in ime and space ha a single ehicle is hi
h oughou he yea , mul iplied by he hou ly a ic and he
annual numbe o hou s o which his a ic condi ion is
alid, esul ing ha
P(S:T)i
is unc ion o he a e age speed
and he size o he ehicle. The p obabili y ha he impac
wi h in as uc u es, he isk quan i ica ion should be usu-
ally ela ed o he a e age consequences in e ms o socie al
isk, ha gi es a isk alue o a whole a ea, no ma e p e-
cisely whe e he ha m occu s wi hin ha a ea, being unc-
ion o he equency o each haza d (Melzne e al. 2020;
De Biagi 2017), he o al numbe o people a ec ed and
hei exposu e (Jonkman e al. 2003; Muhlbaue 2004). As
o en equi ed by policy-make s, he socie al isk is calcu-
la ed in e ms o annual p obabili y o ha ing a leas one
damage ( a ali y), app oxima ed o he numbe o damages
pe yea . Fo elemen s a isk wi h non uni a y exposu e,
e.g. mo ing ehicles, he Au ho s ha e p oposed a me hod
o quan i y he a al and inju y- ela ed consequences o a
block eaching he oad, based on an e en - ee app oach
(ETA) (Ma chelli 2020; Ma chelli e al. 2022a), which is
commonly used o isk assessmen along anspo a ion
co ido s (Bunce e al. 1997a; Mignelli e al. 2012; Mineo
2020; Maccio a e al. 2016a). In his me hod, he occu -
ence p obabili y o a ock all e en is conside ed in bo h
spa ial and empo al dimensions.
The ETA app oaches a e oo ed om he occu ence
o an e en , s a ing om which all he possible scena ios
and ou comes a e e alua ed. He e, he elemen a isk is he
ehicle wi h a elle s inside. The a i al on a block on he
oad is he ini ia ing e en , ollowing which wo mu ually
exclusi e scena ios can de elop: he block can hi he ele-
men a isk o no . In he la e case, he block can ebound
on oad pa emen , e en damaging i s su ace o s op on i .
Each scena io could lead o a ali y o inju y, acco ding o
se e al pa ame e s, ela ed o exposu e (i.e. he speed and
he size o he ehicle, he numbe o people inside) and ele-
men ’s ea u es (i.e. he a io be ween he decision and he
s opping sigh dis ances, and he a ic condi ions), and i
is cha ac e ised by a p obabili y. The p obabili y o ad e se
ou come (say, ic im) is ob ained as he summa ion o he
Fig. 1 P oposed me hod o isk assessmen o people d i ing in a oad subjec ed o ock all, modi ied a e Ma chelli e al. (2022a)
1 3
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Bulle in o Enginee ing Geology and he En i onmen (2025) 84:471
o blocks on he oad. Each sec ion is u he di ided in o
po ions, which a e homogeneous in e ms o a ic condi-
ions. Each po ion is hen subdi ided in o sub-po ions, each
de ined by a uni o m eaching p obabili y.
Thanks o he p ope ies o ETA and he modula i y o
he me hod, he isk can be compu ed o each po ion and
hen agg ega ed ac oss sec ions and, ul ima ely, along he
en i e oad.
I is wo h men ioning ha he exposu e model adop ed
in his s udy is designed o be modula and adap able. While
a simpli ied ep esen a ion is used o gene al applicabili y,
he model can inco po a e eal-wo ld a ic a iabili y, such
as peak-hou a ic, seasonal closu es, o ime-dependen
low a es. Risk calcula ions can be pe o med o sub-annual
in e als wi h consis en a ic condi ions, and he esul ing
p obabili ies can be summed o es ima e he o al annual isk.
This app oach main ains he in eg i y o he e en - ee ame-
wo k while allowing o di e en a ic condi ions. I should
be no ed, howe e , ha he cu en model does no accoun
o seconda y ehicle inciden s, such as collisions o skidding
caused by e asi e maneu e s du ing a ock all e en .
Applying all hese concep , in he amewo k o a di -
used haza d, named
NB,p
he annual equency o e en s
eco ded in he p h sec ion, he es ima ed annual equency
on he k h po ion o he p h sec ion is compu ed acco d-
ing o he each p obabili ies on he pa h ob ained om he
ock all ajec o y analyses, esul ing in:
N
B,k =
Ω
k
∑
n
j=1
Ωk
NB,p
,
(6)
whe e
Ωk
de i es om he weigh ed sum o he each p ob-
abili ies
P(S:B)
and he leng hs o he sub-po ions cons i-
u ing he k h po ion, i.e.:
Ω
k=
∑
o=1
Po
(S:B)ℓo
.
(7)
The summa ion o Eq. (6) is ex ended hus o he o al num-
be (n) o po ions in he sec ion. I is wo h no ing ha he
edis ibu ion o he equencies on he po ions necessa -
ily depends on he p opaga ion analyses, and he choices
behind hei e alua ion, i.e. he elease zones, he block-soil
in e ac ion.
leads o a a ali y,
Pi, a al
, depends also on he numbe o
a elle s. I es ing condi ions apply, e.g. in case o a pa k-
ing, he s op du a ion mus be conside ed in he e alua ion
o
P(T:V)i
. The same p ocedu e is adop ed o he o he
wo scena ios wi h possible a ali ies. The p obabili y o
blocks s opped on he oad
P
o ebounding
(1 −P )
can
be compu ed h ough ajec o y analyses, while he p ob-
abili y o damages on he oad
P
de i es on s a is ics o
pas obse ed e en s. The empo al spa ial p obabili y ha
a ehicle impac s agains a s opped block is unc ion o he
decision sigh dis ance and he speed o he ehicle. Consid-
e ing no a ions epo ed in Fig. 1, he p obabili ies
Ps, a al
and
Pd, a al
can be compu ed om he annual s a is ics o
a al acciden o simila ype o oad and causes. I esul s:
PFi=P(S:T)iPi, a al,
(3)
PFs=[1−
P
(S:T)i]
P
P
(S:T)d
P
s, a al,
(4)
and
PF
d
=[1−
P
(
S
:
T
)i](1 −
P
)
P P
(
S
:
T
)d
Pd, a al
.
(5)
Inse ing Eqs. (3), (4) and (5) in Eq. (1), he p obabili y o
ha ing a a ali y in he k h po ion is ob ained, p o ided he
occu ence o an e en .
In such si ua ions in which he haza d is di used along he
oad, he in o ma ion on pas e en s is ypically ela ed only
o he phenomena which ha e eached he oad. The eco ded
da a can be binned in he a ious pa s o he ack. The leng h
o each pa depends on he su ounding en i onmen and
i s bounds a e de e mined based on he analysis o he ock
aces and slopes. C i ical spo s along he pa h, i.e. o e hang-
ing cli s o limi ed leng h ha o en elease ock blocks, a e
conside ed as speci ic sec ions. Th ough p opaga ion analy-
sis, he me hod also allows o he e alua ion o he spa ial
p obabili ies associa ed wi h di e en blocks eaching he
oad, e e ed o as eaching p obabili y. Mo eo e , a ying
a ic condi ions may occu along he same oad segmen .
To add ess hese complexi ies, he oad is disc e ized in o (i)
sec ions, (ii) po ions, and (iii) sub-po ions. As illus a ed
in Fig. 2, he en i e s e ch om A o D is di ided in o sec-
ions, each cha ac e ized by simila ock all condi ions, such
as he same sou ce a ea and/o compa able geomechanical
p ope ies o he slope ace, esul ing in a simila equency
Fig. 2 Ske ch o a oad wi h i s
disc e iza ion. Sec ions a e ma ked
wi h hick black lines, po ions
wi h ed lines, and sub-po ions
wi h blue lines
1 3
471 Page 6 o 21
Bulle in o Enginee ing Geology and he En i onmen (2025) 84:471
unce ain ies. To his end, a hyb id quan i a i e me hodology
is de eloped, ailo ed o la ge-scale ock all isk assess-
men s in da a-sca ce en i onmen s. This app oach com-
bines elemen s o likelihood-based easoning, uzzy logic,
and Bayesian in e ence o e ec i ely cap u e epis emic
unce ain y. Concu en ly, alea o y unce ain y is managed
h ough a p obabilis ic amewo k employing Mon e Ca lo
simula ions, pa icula ly du ing he ock all p opaga ion
phase, enabling obus es ima ion o impac p obabili ies
and spa ial haza d indica o s. In he ollowing he comple e
QRA me hod is p esen ed, ecalling all he s eps o he isk
assessmen .
Fi s , he sou ce a eas we e de ined based on he geo-
logical in o ma ion a ailable, wi h pa icula emphasis
on p e ious documen ed e en s and he (e en ual) s ud-
ies on he slopes insis ing on he oad. Gi en he na u e o
he analysis, which en isages an isk in eg a e along he
en i e oad, he de ailed geomechanical analysis consis ing
in he iden i ica ion o he discon inui y se s, he elease
mechanisms, and he in-si u block sizes dis ibu ion we e
pe o med o a limi ed numbe o cli s selec ed as ep-
esen a i e o he whole condi ions. The iden i ied sou ce
a eas we e epo ed in a GIS en i onmen . Then, he whole
oad was di ided in o sec ions, acco ding o he de ini-
ion p e iously s a ed, which a e cha ac e ised by simila
geomechanical condi ions o he ace and geomo phology
o he slope, esul ing in a homogeneous po en ial elease
equency. The second s ep consis ed in he de ini ion o
he blocks a i al equencies and he ela ed unce ain ies.
The equency o occu ence o ock all blocks eaching,
o example, he p h sec ion,
NB,p
, was es ima ed based
on he knowledge o p e ious documen ed e en s and he
in o ma ion ob ained om non-documen ed e en s such as:
damages o oad pa emen , blocks in e cep ed by he eg-
e a ion in he su oundings o he oad, blocks in e cep ed
by p o ec i e s uc u es (ne ences), e c. As he ope a ion
o calcula ing he equency o ock alls is a ec ed by epis-
emic unce ain y, he Au ho s conside ed wo addi ional
equencies o occu ence o each sec ion, i.e.
N−
B,p
and
N+
B,p
. The o me was ob ained di iding
NB,p
imes 1.5 o
conside ha he p ocess is less equen han expec ed, he
la e mul iplying he igu e imes 1.5 o conside ha he
p ocess is mo e equen han expec ed. A p obabili y (like
a p io p obabili y) is associa ed o he h ee equencies o
measu e he likelihood o he assump ion. In o he wo ds,
LB,p
is he likelihood associa ed o
NB,p
, i.e. he p ob-
abili y ha he annual equency o occu ence in he p h
sec ion is
NB,p
. The sum
L−
B,p +L+
B,p
, i.e. he sum o he
likelihoods associa ed o
N−
B,p
and
N+
B,p
, is he unce ain y
in he es ima ion o he equency. The sum o he h ee like
likelihoods is one:
The esul o Eq. (6) can be used o map o each po ion
o he oad he p obabili y ha a leas a block a i es du -
ing a
τ
yea s, namely
P(S:T)k(
τ
)
. Conside ing he ock all
occu ence p ocess as a Poisson poin p ocess (McClung
1999; Han z e al. 2003; S aub and Schube 2008; Al-
Shaa e al. 2024), i esul s:
P(
S
:
T
)
k
(τ)=1−exp (−N
B,k
τ).
(8)
I
τ=1
, Equa ion. (8) p o ides he annual p obabili y ha
he k h po ion o oad is a ec ed by a leas one ock all
e en . The hypo hesis ha ock all occu ences on he oad
ollow a Poisson poin p ocess is suppo ed, as he consid-
e ed equency pe ains only o blocks ha ha e eached he
oad. In his amewo k, i mul iple blocks each he ele-
men a isk du ing a single ock all e en , he e en is s ill
ea ed as a single occu ence. Acco ding o he de ini ions
p o ided by Han z e al. (2021), his ype o occu ence can
be e e ed o as he “e en passage equency” on he oad,
which ela es speci ically o he occu ence o a ock all
e en . I he equency o ock alls on he sec ion is low, he
p obabili y compu ed wi h Eq. (8) p o ides he p obabili y
o ha ing a single ock all e en in he yea on he oad po -
ion, and he annual p obabili y o ha ing a a ali y, i.e. he
isk, on he p h sec ion can be compu ed as:
R
p=
n
∑
k=1 {
[1 −exp (−NB,k)] PFk
},
(9)
whe e he summa ion ex ends o all he n po ions cons i u -
ing he p h sec ion. The isk on he en i e oad is compu ed
as he sum o he alues o each sec ion.
Equa ion (9) unde es ima es he eal isk i he hypo h-
esis o a low e en s’ equency on he sec ion does no hold.
Hence, a mo e gene al app oach based on Binomial dis i-
bu ion is conside ed. The annual p obabili y o ha ing a
leas one ic im, ha is he isk, on he p h sec ion is p o-
ided by:
R
p=
n
∑
k=1 [
1−
(
1−PFk
)
NB,k
],
(10)
being he e m in squa e b acke s he isk in each po ion.
Me hodology
The a iables in ol ed in he p e iously desc ibed app oach
o quan i ying he socie al isk along a oad a e ea ed o
add ess bo h alea o y and, mo e impo an ly, epis emic
1 3
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Bulle in o Enginee ing Geology and he En i onmen (2025) 84:471
likelihood associa ed was he p oduc o he single likeli-
hoods. This allowed o associa e a con idence alue o he
quan i ied isk, as p esen ed in Sec ion “Case s udy”.
Case s udy
The me hodology desc ibed in Sec ion “Me hodology” is
applied o a case s udy in Valpelline alley in Aos a Val-
ley, No hwes e n I alian Alps. Valpelline is a 33 km long
alley ha s a s 6.5 km no h o Aos a and uns NE-SW
pa allel o he bo de be ween I aly and Swi ze land (Wal-
lis can on). Figu e 3 ske ches he a ea and he geog aphical
con ex . A abou h ee-qua e s o he leng h o he alley
he Places Moulin dam, a 136 m high double-cu a u e ein-
o ced conc e e dam, is loca ed. The dam o ms an a i icial
lake loca ed a 1965 m a.s.l. and 6.5 km long wi h a capaci y
o 105 Mm
3
ha se es as s o age o p ecipi a ions and
mel wa e om he glacie s ha a e p esen in he uppe
pa o he alley and low along Bu hie c eek. A sealed
oad s e ching om Bionaz illage o he dam and buil
in he Six ies (in yellow in Fig. 3) is open om Ap il o
Oc obe and closed in he win e pe iod due o snow a a-
lanche haza d. The accessibili y o he dam du ing he win-
e season is gua an eed ia helicop e . The dam si e and
he lake a e e y popula among ou is s, and he pa king
a ea loca ed close o i is he s a ing poin o se e al hikes
and moun ain ails. The 7.5 km long oad is he only ou e
o access o he dam and he high pa o Valpelline alley.
In 2023, Bionaz municipali y has coun ed mo e han 13k
ehicles in he paying pa king lo s. Th ee ypes o ehicles
access he oad: ca s (88%), coaches (2%) and mo ocycles
(10%) wi h an es ima ed a ic o 150.2 ehicles pe day
(bo h ways) in he opening season conside ing ha some o
he p e ious ca ego ies do no pay he pa king and ha he e
a e ee pa king as well. The a e age speed along he oad
is 50 km/h.
Besides snow a alanches and o he landslide haza ds
(deb is low in he gullies and looding along he c eeks),
he oad is also subjec ed o ock alls o (almos ) i s en i e
leng h om Bionaz illage (Bi in Fig. 3). F om p e ious
gene al s udies on he si e de o ed o land use planning,
i esul s ha a o al o 6310 m, i.e. om s o e in Fig. 3,
a e po en ially subjec ed o he haza d. Al hough a de ailed
desc ip ion o he geomo phological con ex is ou lined in
Sec ion “Geological knowledge and p e ious e en s”, i is
an icipa ed ha he a eas ups eam he oad p esen a ie-
ga e ypes o soil and ege a ion (loose ock, deposi , g az-
ing, o es ), and di e en slope angles, om gen le slope
o o e hanging cli s. Figu e 4 shows he di e en condi-
ions along he oad. E en i ock all isk mi iga ion mea-
su es we e ins alled along he oad since i s opening (ne
L−
B,p +L
B,p
+L+
B,p =1.
(11)
Based on he a ious soil ypes, back-analysis o p e ious
e en s, and a ailable li e a u e on simila en i onmen s,
he p opaga ion analyses we e conduc ed wi hin a p oba-
bilis ic amewo k using Mon e Ca lo sampling echniques.
This allowed o he de e mina ion o eaching p obabili ies
and he subdi ision o he a ea in o dis inc sub-po ions.
Finally, ealis ic a ic scena ios we e de ined and he cal-
cula ions o he isk acco ding o he o mulae p esen ed
in Sec ion “Basics o isk calcula ion on oads” we e pe -
o med. In he p esen case, he conside ed elemen s a isk
a e people d i ing on a oad. In his con ex , due o high
eloci ies in ol ed in a ock all e en (up o 30 m/s) and o
he kine ic ene gies o he possible impac ing blocks, i was
assumed ha any block o any size could cause damage.
This assump ion is suppo ed by mul iple s udies indica -
ing ha e en small blocks can lead o se e e consequences.
Hoek and Ka akas (2008) ha e epo ed ha ock alls ha e
caused nume ous a ali ies on moun ain highways, wi h
small blocks capable o p oducing deadly impac s due o
high eloci ies and slope geome ies, such as ski-jump
e ec s. Simila ly, Rwodzi (2010) has highligh ed ha small
blocks can pene a e windshields o des abilize ehicles,
pa icula ly a highway speeds. Ma ouli and Co ominas
(2018) and Maheshwa i e al. (2023) ha e u he empha-
sized ha any ock wi h su icien kine ic ene gy o damage
pa emen can also inju e o kill ehicle occupan s. These
indings a e ein o ced by ehicle sa e y da a om he
Na ional Highway T a ic Sa e y Adminis a ion (NHTSA),
which has indica ed ha occupan inju ies can occu a
impac speeds as low as 4–5 mph (6–8 km/h) (Nama e al.
2016), especially in low-c ush scena ios whe e ene gy is
ans e ed di ec ly o passenge s.
Th ee di e en alues o isk, in e ms o annual p ob-
abili y o ha ing a leas one a ali y, we e ob ained o each
sec ion, each o which is associa ed wi h likelihood alue,
based occu ence equency adop ed in he calcula ion.
Equa ion (10) was a anged o include he likelihood alue
on he occu ence equency:
R
p=
3
∑
ℓ=1
Lℓ
B,p
{
n
∑
k=1 [
1−
(
1−PFk
)
Nℓ
B,k
]}.
(12)
I is wo h o no e ha he app oach holds o any numbe
o occu ence equencies (hence, no necessa ily 3 as done
by he Au ho s), p o ided ha he co esponding likeli-
hoods espec Eq. (11). Conside ing ha he oad is made o
a o al o s sec ions,
3s
combina ions o he isk alues we e
e alua ed. Fo each combina ion, he isk along he en i e
oad was compu ed as he sum o he single isks, and he
1 3
471 Page 8 o 21
Bulle in o Enginee ing Geology and he En i onmen (2025) 84:471
p o ec ion measu es is no a ailable, as well as he knowl-
edge o hei e iciency and e ec i eness. On he con a y, a
ew documen al in o ma ion is a ailable on he ecen p o-
ec ion wo ks.
The app oxima e size o he s udy a ea is 5.3 km long and
app oxima i ely 1.5 km wide, i.e. 7.95 km
2
o su ace, wi h
an ele a ion anging om 1650 m o 2750 m a.s.l.
ences and d ape y meshes), some o hem ins alled in he
las 10-15 yea s. F om a de ailed su ey, i esul s ha some
o he ne ences a e comple ely us ed and unable o s op
blocks, anymo e. Se e al ne ences we e o e u ned by he
p essu e exe ed by snow a alanches and, among he d ap-
e y meshes, local ea ing and lace a ions a e isible. In gen-
e al, he in o ma ion on he o iginal design o he ock all
Fig. 4 Th ee ae ial iews o he oad and he upslope cli s
Fig. 3 Ske ch o he op pa o
Valpelline alley (Aos a Valley).
The oad ha uns along he alley
and links Bionaz (Bi) wi h Places-
Moulin lake (P) is in yellow. The
pa o oad subjec ed o ock alls
is be ween S and E ed ma ks. The
majo idges and c eeks a e ma ked
in black and blue, espec i ely. The
scale ma ks he kilome e s. The
map o I aly in he bo om igh
highligh s he posi ion o Aos a
Valley
1 3
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Bulle in o Enginee ing Geology and he En i onmen (2025) 84:471
The map o he each p obabili y,
P(S:B)
, shown in
Fig. 6.a, is use ul o managing isk. I helps iden i y which
sou ce a eas impac longe s e ches o oad and, he e o e,
may pose a g ea e isk o exposed elemen s. This in o ma-
ion can be used o de e mine whe e p e en i e measu es
should be p io i ised o checking i he cu en p o ec ion
s uc u es a e loca ed in he op imal posi ion. Figu e 6.b, on
he o he hand, p esen s he annual p obabili y o a leas
one occu ence. This map complemen s ha in Fig. 6.a, as
i highligh s he mos c i ical zones whe e blocks a e mo e
likely o a i e, inco po a ing empo al in o ma ion. As a
esul , he alues shown may no necessa ily coincide wi h
hose in he each p obabili y map. This map suppo s he
iden i ica ion o a eas whe e p o ec i e measu es should be
p io i ised. Na u ally, his map can only be p oduced when
a ca alogue o pas e en s is a ailable.
Analysing hen he sou ces o unce ain y, i should be
no iced ha he co e o he p ocedu e o isk quan i ica ion
lies on he de ini ion o he numbe o e en s along he in a-
s uc u e. Hence, he app oach implici ly accoun s o unce -
ain ies in all p ocesses along he slope, om de achmen o
p opaga ion, including po en ial agmen a ion. The epis-
emic unce ain y in he de ini ion o he numbe o e en s
has been sol ed by in oducing a likelihood o ha ing a ce -
ain equency o occu ence. To assess how e ec i ely he
me hod cap u es unce ain ies in es ima ing he occu ence
R co esponding o 50- and 90-pe cen iles a e epo ed in
Table 6.
Discussions
The analysis indica es ha he socie al isk, compu ed
as he p obabili y o ha ing a leas one a ali y pe yea ,
along he oad is smalle han
1×10−5
. This alue has
been compu ed conside ing ha only he e ec i e ne
ences a e able o in e cep and s op he alling blocks,
while he old ba ie s a e neglec ed. Resul s in e ms o
haza d and isk a e bo h use ul o de ine managemen
s a egies, o iden i y he po ions o he e y la ge a ea
ha need de ailed analyses, and, e en ually, o design and
ins all addi ional mi iga ion measu es. Fi s , he ob ained
isk alues should be compa ed o an accep able h eshold
and, i highe , isk educ ion plans should be e alua ed.
Fo socie al isk, he I alian S anda d UNI 11211-2 (2021)
sugges s o conside a alue in be ween
10−6
and
10−5
pe
yea as easonably accep able in case o ock all p o ec ed
s uc u es o in as uc u es. Howe e , i mus be no ed
ha he alue o such h eshold is s ill unde deba e in he
scien i ic communi y (En igh 2015; Sim e al. 2022). To
he knowledge o he Au ho s, a p ede ined alue has no
been de ined ye .
Table 6 Values o R co esponding o 50- and 90-pe cen iles o he di e en a ic scena ios analyzed
T a ic (pe y )
R|P( ≤R)=0.5
R|P( ≤R)=0.9
32k ehicles
4.68 ×10−6
5.36 ×10−6
40k ehicles
5.86 ×10−6
6.66 ×10−6
50k ehicles
7.32 ×10−6
8.23 ×10−6
60k ehicles
8.78 ×10−6
9.97 ×10−6
00.2 0.40.6 0.811.21.4 1.
6
Risk R 10
-5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
P ( <R)
32k
40k
50k
60k
T a ic (pe yea )
Fig. 7 Risk s. cumula i e likeli-
hood cu es o di e en annual
a ic condi ions. The con inuous
lines e e o he isk compu ed
conside ing he inpu da a o
Table 3, while dashed cu es ela e
o he 90% con idence bounds, as
de ailed in Sec ion “Discussions”
1 3
471 Page 16 o 21
Bulle in o Enginee ing Geology and he En i onmen (2025) 84:471
blocks accumula ed behind he ba ie we e added o he lis
o blocks ha eached he oad in a 10 yea s pe iod. The
upda ed equencies
NB,p
is 0.50, 0.70, 0.50, 0.60, o Sec-
ions “In oduc ion”, “Me hodology”, “Discussions” and 7,
espec i ely and he co esponding isks a e epo ed in he
igh -hand side column o Table 7. He e, a uni likelihoods
is associa ed o each equency. These isks e e o he case
in which main enance o he ba ie s is no longe pe o med
and he cu en mi iga ion measu es unde go ageing and
se e e deg ada ion.
Re e ing o he ob ained likelihood, looking a o he
me hods, such igu e can be in ended as a p io dis ibu-
ion in a Bayesian in e ence analysis. In uzzy isk analy-
sis, a p obabili y is a ibu ed o he easibili y o a alue
o an inpu a iable. The app oach is simila o he one
p oposed by he Au ho s. I is wo h no ing ha he likeli-
hood associa ed o he possible occu ence equencies ha
can be assigned o a speci ic oad sec ion can esul om
independen e alua ions o a pool o expe s. Fo example,
each expe could be asked o indica e wha hey hink is he
likely equency o occu ence ( om a ange o p ede ined
equencies). The esul s o he su ey would p o ide he
likelihoods o be inco po a ed in o he isk analysis. How-
e e , i is impo an o emind ha missing e en s can ha e
ele an consequences when he equency o occu ence
is aken as inpu o addi ional analyses (De Biagi 2017):
hence, a en ion mus be p o ided o he e alua ion o his
pa ame e .
Implica ions and ecommenda ion
The esul s o he analyses p o ide a cohe en and quan i a-
i e amewo k o unde s anding he ock all haza d a ec -
ing he s udy a ea and po en ial consequences, wi h di ec
implica ions o oad managemen and e i o ial planning.
The p esence o nume ous po en ial ock all sou ces, com-
bined wi h he s eep mo phology o he slopes, esul s in
a signi ican spa ial p obabili y o block each in he e en
o de achmen . This condi ion is pa icula ly c i ical in he
absence o de ailed in o ma ion on exis ing p o ec i e s uc-
u es, which can also no be conside ed in he modelling due
o he lack o design documen a ion and he he e ogeneous
s a e o conse a ion obse ed du ing ield inspec ions. The
o he phenomenon, Table 7 also epo s he annual isk al-
ues ob ained by assuming ha he equencies
NB,p
ha e
uni likelihood, i.e. no unce ain ies a e conside ed. Such
alues, named as Q, can be compa ed wi h he minimum
and maximum isks, as well wi h he isks co esponding o
speci ic pe cen iles, say 50 h and 90 h, epo ed in Table 6.
The esul s show ha , i espec i ely om he annual a ic,
he isk compu ed unde his assump ion app oxima ely co -
esponds o he 50 h pe cen ile o he empi ical dis ibu ion
o isk and likelihoods. The ange o he compu ed isks,
i.e.
(max R−min R)
, is p og essi ely inc easing, om
3.48 ×10−6
a 32k ehicles pe yea , o
6.52 ×10−6
a 60k
ehicles pe yea .
The p obabili y o a a al acciden , i.e.
PF
in Eq. (1), p i-
ma ily in luenced by he ehicle’s eloci y and he numbe
o occupan s, can inc ease o dec ease depending on he
a e age speed on he in as uc u e and he a e age num-
be o occupan s pe ehicle. As epo ed in he li e a u e,
ehicle eloci y a ies signi ican ly, especially on moun-
ain oads (Andueza 2000; Nama e al. 2016). To accoun
o pa ame e a iabili y, a coe icien o a ia ion o 0.2
ela i e o he mean alues o he indi idual p obabili y o
a ali y is conside ed. The e o is p opaga ed om he indi-
idual p obabili y o he o e all p obabili y o a ali y,
PFk
o Eq. (10), acco ding o he numbe o ehicles. Figu e 7
p esen s he 90% con idence bounds o he isk-likelihood
cu es. As expec ed, he bounds widen as he numbe o
ehicles inc eases; a
P( <R)=0.5
, he wid hs o he
bounds a e
3.1×10−6
,
3.88 ×10−6
,
4.85 ×10−6
and
5.82 ×10−6
, o 32, 40, 50, and 60 housands ehicles,
espec i ely.
Finally, o assess he isk in he e en ha he ne ences,
p e iously conside ed e ec i e, ail o in e cep and s op
he blocks, he isk was ecalcula ed o he ou a ic sce-
na ios. The numbe o e en s on hose Sec ions in which
p o ec ion sys ems a e ins alled (1, 3, 5 and 7) was de e -
mined based on he su eys pe o med by he Au ho s along
he oad. The conside a ions ela e o he ba ie s ha ha e
been ecen ly ins alled, only, as he p o ec ion p o ided
by he olde and no e icien ones has been disca ded in
he calcula ions. To quan i y he ex a blocks eaching he
oad, i has been conside ed ha he ba ie s ha e been
ins alled, on a e age, 10 yea s ago. This means ha he
Table 7 Risk compu ed conside ing no unce ain ies in he occu ences, Q, and minimum and maximum isks conside ing he unce ain ies, o he
di e en analysed a ic scena ios. The igh -hand side column epo s he isks in case he p o ec i e s uc u es would be no e ec i e
T a ic (pe y ) No unce ain ies, Q
min R
max R
No ne - ences, U
32k ehicles
4.73 ×10−6
2.79 ×10−6
6.27 ×10−6
6.02 ×10−6
40k ehicles
5.91 ×10−6
3.49 ×10−6
7.84 ×10−6
7.51 ×10−6
50k ehicles
7.39 ×10−6
4.36 ×10−6
9.80 ×10−6
9.39 ×10−6
60k ehicles
8.87 ×10−6
5.24 ×10−6
11.76 ×10−6
1.13 ×10−5
1 3
Page 17 o 21 471
Bulle in o Enginee ing Geology and he En i onmen (2025) 84:471
is limi ed knowledge o he a ea. The isk is compu ed ol-
lowing he p ocedu e illus a ed by he Au ho s in Ma chelli
(2020); Ma chelli e al. (2022a), which equi es de ails on
a ic condi ions (numbe o ehicles, speed, e c.) and on he
occu ence o ock all e en s along he oad. The o me poin
can be easily es ima ed by measu ing he cu en a ic, o by
p oposing new scena ios based on he expec ed u ban- ou is-
ic de elopmen o he a ea. The la e , which is di ec ly linked
o he haza dous phenomenon, can be de ined in p obabilis ic
e ms, by conside ing di e en occu ence equencies and
associa ing o each o hem a likelihood, i.e. a p obabili y ha
measu es he deg ee o expec a ion, co esponding o a p io
p obabili y in a Bayesian app oach. The occu ences and he
co esponding likelihoods a e de e mined om he analysis
o he in en o y o pas e en s en iched wi h he in o ma ion
aken om on-si e su ey ( om p o ec ion s uc u es, dam-
aged elemen s, e c.). The ou pu o he p ocess consis s in
a se o alues o isk and associa ed likelihoods. The isk
co esponding o a speci ic pe cen ile se es as basis o isk
managemen . In addi ion, wo maps de ine he a eas in which
de ailed analyses a e needed. The limi a ions o he p ocedu e
and he sou ces o unce ain y ha e been discussed. The p o-
cedu e is sui able o be implemen ed in isk analyses when
limi ed knowledge is p esen , aking ca e o he limi a ions
and he implica ions o he hypo heses done.
Au ho Con ibu ion MM: Concep ualiza ion, Me hodology, Fo mal
analyses, W i ing - o iginal d a , W i ing - Re iew & Edi ing, P ojec
adminis a ion, Funding acquisi ion. VDB: Concep ualiza ion, Me h-
odology, Fo mal analyses, W i ing - o iginal d a , W i ing - Re iew
& Edi ing, P ojec adminis a ion, Funding acquisi ion. MP: Concep-
ualiza ion, Da a acquisi ion, W i ing - Re iew & Edi ing, P ojec ad-
minis a ion. DB: Concep ualiza ion, Da a acquisi ion, P ojec admin-
is a ion.
Funding Open access unding p o ided by Poli ecnico di To ino wi h-
in he CRUI-CARE Ag eemen . This wo k was pa ially suppo ed by
he esea ch p ojec “Se izio di suppo o al amen e specialis ico al Re-
sponsabile Unico del P ocedimen o nell’ambi o della p edisposizione
di s umen i pe la ges ione del ischio da cadu a massi su insediamen i
abi a i i e della alu azione de ischio su s ade ad al a ulne abili á”
unded by Regione Au onoma Valle d’Aos a (I aly) and pa ially by
he Ma ie Cu ie Pos doc o al Fellowship 2022 (Call Ho izon-MSCA-
2022-PF-01, G an GA101103401 (RIDETHERISK p ojec ).
Open Access This a icle is licensed unde a C ea i e Commons
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seasonal inc ease in ehicula and pedes ian a ic, espe-
cially du ing he summe mon hs, u he ampli ies he ul-
ne abili y o he oad in as uc u e and adjacen es a eas.
The mapping o annual eaching p obabili y, which accoun s
o he a iabili y in ock all equency along di e en oad
segmen s, allows o a mo e e ined isk assessmen and
suppo s he p io i isa ion o in e en ion s a egies.
In ligh o hese indings, i is ecommended o imp o e
he geological cha ac e isa ion o he ock masses h ough a -
ge ed ield in es iga ions, including ope-access su eys and
olume ic assessmen s o po en ially uns able blocks, wi h
pa icula a en ion o sou ce a eas ha could a ec ex ensi e
segmen s o he oad, as can be in e ed om Fig. 6.a. Among
he possible mi iga ion s a egies, and speci ically conside -
ing s uc u al measu es, he ins alla ion o ne ences eme ges
as a po en ially p o i able solu ion. These sys ems can be
designed wi h ailo ed ene gy abso p ion capaci y and heigh ,
based on de ailed ajec o y analyses, o e ec i ely in e cep
alling blocks. This app oach is p e e able o secu ed d ape y
meshes and ock bol ing on he po en ial uns able zones, due
o he ex ensi e na u e o he po en ial sou ce a eas, which
limi s he e ec i eness o localised s abilisa ion echniques.
Gi en he a ailable inancial esou ces, he map in eg a ing
bo h he p obabili y o each and he equency o e en s, i.e.
Figu e 6.b, could p o ide a cos -e ec i e basis o p io i iz-
ing a eas o p o ec ion. Conside ing he po en ial block ol-
umes eaching he oad (VRU equal o 1 m
3
), embankmen s,
al hough gene ally eliable o wi hs anding mul iple impac s
and equi ing minimal main enance, may p o e economi-
cally un easible in his con ex . Ne e heless, a sys ema ic
and long- e m e alua ion o he condi ion o all he p o ec i e
(o p e en i e) s uc u es, bo h exis ing and u u e, should be
unde aken o suppo main enance planning and ensu e con-
inued e ec i eness o e ime (Ma chelli e al. 2019, 2020b,
2022b). Fu he mo e, conside ing non-s uc u al measu es,
a ic managemen can ac as a non-s uc u al isk mi iga ion
measu e. The maximum allowable numbe o ehicles pe day
can be ixed, based on isk h eshold. I is also ad isable o
egula e access o he oad du ing pe iods o in ense ain all,
which a e known o inc ease he likelihood o ock all e en s
(Delonca e al. 2014; D’Ama o e al. 2016). Finally, p io o
he seasonal eopening o he oad, inspec ions should be ca -
ied ou o iden i y and emo e any uns able blocks. These
measu es aim o enhance he sa e y and esilience o he in a-
s uc u e while suppo ing he sus ainable use o he a ea.
Conclusions
A speci ic eques om a Public Adminis a ion led o he
design o a new me hodology o quan i y ock all socie al
isk along a 7.5 km oad in he pa icula case in which he e
1 3
471 Page 18 o 21
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