Reliability analysis of beaches as defenses against storm impacts under present and future climate change scenarios

T eball eali za pe :
Blanca Be ià B ey
Di igi pe :
F ancesc Xa ie Gi onella i Cobos
Vicen e G acia Ga cia
G au en:
Enginye ia Ci il
Ba celona, 28 de Gene del 2025.
Depa amen d’Enginye ia Ci il i Ambien al
TREBALL FINAL DE GRAU
Reliabili y Analysis o Beaches as
De ences Agains S o m Impac s
Unde P esen and Fu u e Clima e
Change Scena ios
2
Acknowledgemen s
I would like o exp ess my since e g a i ude o P o esso s F ancesc Xa ie Gi onella i
Cobos and Vicen e G acia Ga cia o hei guidance h oughou his p ojec . They
p o ided excep ional suppo om he beginning, o e ing clea di ec ion, necessa y
esou ces, and aluable men o ship ha we e c ucial o comple ing his wo k
success ully.
I am deeply hank ul o my amily o hei cons an suppo . Thei in e es in my
esea ch and hei commi men in my academic jou ney ha e been essen ial. Thei
encou agemen has helped me o e come challenges h oughou his p ocess.
My expe ience s udying Ci il Enginee ing a UPC has been ans o ma i e. Du ing my
ime he e, I ha e disco e ed he ascina ing aspec s o ci il enginee ing, including i s
a ious specializa ions and p ac ical applica ions. The bachelo has p o ided me wi h
bo h echnical knowledge and impo an pe cep ions abou he ield's impac . I ha e
had he oppo uni y o s udy wi h excellen classma es who ha e become alued
colleagues. Toge he , we ha e spen many hou s wo king on p ojec s, engaging in
echnical discussions, and suppo ing each o he . These mu ual aids ha e s eng hened
my unde s anding o eamwo k and c oss-disciplina y coope a ion, c ea ing a s ong
ounda ion o my u u e ca ee in enginee ing.
3
Index
Acknowledgemen s .................................................................................................. 2
Abs ac ................................................................................................................... 5
1. In oduc ion ................................................................................................................. 7
1.1. Mo i a ion ............................................................................................................. 9
1.2. Objec i es ............................................................................................................ 10
1.3. S uc u e ............................................................................................................. 12
2. Theo e ical F amewo k .............................................................................................. 12
2.1. The Eb o Del a: Coas al Cha ac e is ics and En i onmen al Challenges ............ 13
2.2. Coas al looding on beaches: e iew o exis ing equa ions o wa e un-up and
looding on beaches ....................................................................................................... 17
2.2.1. Hun (1959) ............................................................................................... 18
2.2.2. Yamamo o (1972) ..................................................................................... 18
2.2.3. Holman (1986) .......................................................................................... 19
2.2.4. Kobayashi and Tamu a (1986) .................................................................. 19
2.2.5. Mase (1989) .............................................................................................. 20
2.2.6. Nielsen & Hanslow (1991) ........................................................................ 20
2.2.7. Van de Mee & S am (1992) .................................................................... 20
2.2.8. S ockdon e al. (2006) ............................................................................... 21
2.2.9. Flick and Mu phey (2008) ......................................................................... 21
2.2.10. Mase and Shibayama (2009) .................................................................. 22
2.2.11. Summa y o wa e un-up equa ions ...................................................... 22
2.3. Selec ing he App op ia e Wa e Run-up Equa ion o he Eb o Del a ............... 23
2.3.1.The I iba en numbe ................................................................................ 24
2.4. Sea Le el Rise P ojec ions and Regional Impac Analysis ................................... 26
2.4.1 Global Con ex and Medi e anean Speci ici y ......................................... 26
2.4.2. Regional Downscaling and Medi e anean P ojec ions ........................... 26
2.4.3. Tempo al Analysis o P ojec ed Sea Le el Rise ......................................... 26
2.4.3.1. Sea le els p ojec ions ...................................................................... 27
2.4.4. Compa a i e Analysis o P ojec ion Sou ces ............................................ 27
2.4.5. Sea Le el Rise Implica ions and Unce ain ies in Coas al Managemen .. 27
4
2.5. Failu e ee: Hie a chical Analysis o Beach Flooding Componen s and Risk
Assessmen ..................................................................................................................... 29
3. Code Me hodology .................................................................................................... 32
3.1. In oduc ion ........................................................................................................ 32
3.1.1. Mon e Ca lo Simula ion Model o Coas al Flooding Risk Assessmen ..... 32
3.1.2. As onomical Tide Modelling ..................................................................... 33
3.1.3. Me eo ological esidue............................................................................... 35
3.1.4. Wa e Heigh Analysis and Weibull Dis ibu ion ......................................... 37
3.1.5. Di ec ional Wa e Analysis unde S o m Condi ions .................................. 40
3.1.6. Wa e Run-up Calcula ion (S ockdon e al., 2006) ...................................... 42
3.1.7. Sea Le el Rise Scena ios ............................................................................. 45
3.1.8. To al Wa e Le el and Failu e Analysis ....................................................... 47
3.1.9. Con e gence Analysis o Mon e Ca lo Simula ion Resul s ......................... 48
3.1.10. Reliabili y Analysis o S o m-Induced Coas al Flooding .......................... 48
3.1.11. Applica ion o he model: The Eb o Del a Sys em ................................... 52
3.1.12. Beach Cha ac e is ics: Si e-Speci ic Analysis o The Ma quesa Beach .. 54
3.1.13. Beach Cha ac e is ics: Si e-Speci ic Analysis o The T abucado Beach
...................................................................................................................58
3.1.14. Me hodology o Mon e Ca lo Simula ion Va iable Conse a ion .......... 61
3.2.Theo e ical Cons ain s and Assump ions ........................................................... 63
4. Resul s ........................................................................................................................ 64
4.1. The Ma quesa ..................................................................................................... 65
4.1.1. Wa e analysis ............................................................................................. 65
4.1.2. Wa e Le el Componen s ........................................................................... 66
4.1.3. Failu e P obabili y du ing S o m E en s ..................................................... 67
4.1.4. Wa e Le el Exceedance ............................................................................. 68
4.1.5. Con e gence Analysis ................................................................................. 68
4.1.6. Reliabili y Analysis ...................................................................................... 74
4.2. The T abucado ................................................................................................... 77
4.2.1. Wa e Analysis ............................................................................................. 77
4.2.2. Wa e Le el Componen s ........................................................................... 78
4.2.3. Failu e P obabili y Du ing S o m E en s .................................................... 79
5
4.2.4. Wa e Le el Exceedance ............................................................................. 80
4.2.5. Con e gence Analysis ................................................................................. 81
4.2.6 Reliabili y Analysis ....................................................................................... 86
5. Conclusions ................................................................................................................ 88
6. Sus ainabili y analysis and e hical implica ions ....................................................... 90
7. Sou ces ....................................................................................................................... 93
Appendices ..................................................................................................................... 97
1. Model code ............................................................................................................ 97
2. Ma lab used unc ions ......................................................................................... 103

6
Abs ac
This esea ch e alua es he eliabili y o na u al coas al p o ec ion sys ems agains
looding unde clima e change condi ions. The s udy ocuses on wo mo phologically
dis inc beaches in Spain's Eb o Del a: The Ma quesa and The T abucado . I assesses
hei p o ec i e capaci y and eliabili y du ing s o m e en s by analysing how hei
unique geome ic cha ac e is ics in luence hei e ec i eness as lood ba ie s.
The me hodology implemen s a p obabilis ic amewo k using Mon e Ca lo simula ion.
This app oach in eg a es as onomical ides, me eo ological condi ions, and wa e
dynamics o assess s o m impac scena ios. The analysis spans mul iple ime ho izons
(2025, 2050, and 2100) ac oss a ious clima e change scena ios (Cu en , RCP4.5, and
RCP8.5), and includes he e ec o land subsidence in he Eb o Del a, e alua ing bo h
immedia e and long- e m coas al p o ec ion capabili ies.
The compa a i e analysis e eals dis inc sea-le el ise esponse pa e ns be ween he
si es. The Ma quesa beach cu en ly con ibu es o e ec i e coas al p o ec ion a highe
ele a ions (2.2m), hough his eliabili y diminishes unde p ojec ed clima e change
scena ios. The T abucado demons a es g ea e sensi i i y o en i onmen al
condi ions, wi h i s p o ec i e unc ion (2.1m) showing accele a ed de e io a ion. Bo h
loca ions exhibi signi ican ulne abili y in hei lowe sec ions (1.3m and 1.2m
espec i ely), wi h analysis indica ing subs an ial deg ada ion o p o ec i e capaci y by
2100, pa icula ly unde RCP8.5 scena ios.
The esea ch p o ides c ucial quan i a i e e idence o coas al de ence planning by
emphasizing ha p ese ing beach geome ic cha ac e is ics h ough main enance
p og ams is essen ial o sus ained p o ec i e capaci y, ac ion pa icula ly c i ical as
clima e change con inues o eshape coas lines and in ensi y s o m impac s.
Keywo ds: coas al looding, beach eliabili y, clima e change, sea le el ise, s o m
impac , Mon e Ca lo simula ion, Eb o Del a, coas al p o ec ion.
7
1. In oduc ion
Clima e change is p o oundly ans o ming ou wo ld, i e e sibly al e ing ecosys ems,
lo a, and auna. One o i s mos de as a ing e ec s is he ise in sea le els, which is
al eady causing mo e equen and se e e looding in coas al a eas. Combined wi h he
in ensi ica ion o s o ms and hu icanes, hese phenomena a e h ea ening bo h
communi ies and coas al biodi e si y, leading o an unp eceden ed en i onmen al and
social c isis.
As global empe a u es ise and wea he pa e ns shi , coas al communi ies a e acing
a signi ican ba le agains he sea. Coas al looding e en s a e becoming mo e common,
as s o ms g ow in equency and in ensi y. This imminen c isis h ea ens li es, jobs, and
he exis ence o many coas al communi ies, he dange o inunda ion is imminen , and
he ime o ac ion is now. I is necessa y o de elop s a egies o s eng hen he na u al
de ences ha ha e long been p o ec ed in he pas om he o ces o he sea.
Beaches play a c ucial ole in his de ence by abso bing he ene gy o c ashing wa es
and p o ec ing inland a eas om he ull impac o s o m su ges. Thei e ec i eness
depends on wo key ac o s: beach wid h and ele a ion. When hese a e comp omised,
wa es can easily su pass he sho eline, leading o des uc i e coas al looding. Howe e ,
as clima e change p og essi ely e odes hese na u al ba ie s, he ques ion a ises: how
much p o ec ion can he beaches uly p o ide in he ace o u u e s o ms?
The Eb o Del a, loca ed on Spain's Medi e anean coas , s ands as a egion o immense
ecological, economic, and social impo ance whe e he Eb o Ri e mee s he sea (Figu e
1). I is home o ex ensi e ag icul u al lands, we lands, and lagoons ha a e i al o he
local economy and ecosys em.
Figu e 1: Map o he Eb o Ri e Basin in Spain. Sou ce: Locken, The challenge o managing he Eb o Ri e
Basin: La ge han hal he coun ies in he EU. iAgua, h ps://www.iagua.es.
8
Howe e , he del a exempli ies he ulne abili y o coas al a eas, as looding could
inunda e hese egions, causing se e e damage o public in as uc u e and habi a s.
The beaches he e dissipa e wa e ene gy and p o ec ing inland a eas om looding. Ye ,
he excep ional challenges aced by he Eb o Del a make he ole o hese beaches e en
mo e c i ical. Wi h ising sea le els and inc easingly in ense s o ms, he egion's eliance
on hese na u al de ences is apidly educing, lea ing i s communi ies and ecosys ems
mo e ulne able han e e .
This s udy aims o conduc a eliabili y analysis o sandy beach a che ypes along he Eb o
Del a coas line, e alua ing hei e ec i eness as na u al de ences agains s o m
impac s. The analysis will use Mon e Ca lo simula ions o accoun o he a iabili y in
key hyd odynamic and mo phological pa ame e s. By modelling a wide ange o
po en ial scena ios, he s udy seeks o assess he likelihood o beach ailu e and he
associa ed isk o coas al looding unde bo h cu en and u u e clima e condi ions. This
s ochas ic app oach, powe ed by Mon e Ca lo simula ions, o e s a mo e ealis ic
ep esen a ion o he a iabili y and unce ain y inhe en in coas al p ocesses, in
con as o adi ional de e minis ic me hods.
The signi icance o his esea ch becomes e en mo e c i ical in he con ex o clima e
change. As sea le els ise, s o m pa e ns shi , and s o m su ges g ow mo e in ense, he
p essu e on coas al de ences is expec ed o inc ease. This, in u n, accele a es beach
e osion and diminishes hei abili y o p o ec inland a eas. As hese na u al de ences
weaken, he isk o se e e looding and i s economic and humani a ian impac s will
in ensi y. Th ough his s udy, he aim is o p esen undamen al insigh s in o he long-
e m eliabili y o beaches as na u al ba ie s.
The esul s o his esea ch will p o ide essen ial da a o suppo e o s in coas al
p o ec ion and disas e isk educ ion. As clima e change con inues o eshape
coas lines, ensu ing he esilience o na u al de ences, such as beaches, is becoming
e e mo e c ucial o p o ec coas al communi ies, in as uc u e, and economies.
The e o e, his s udy ep esen s a signi ican ad ancemen in unde s anding and
add essing hese challenges, o e ing in e p e a ions ha can help shape e ec i e
s a egies o inc ease coas al esilience in he ace o a changing clima e.
9
1.1. Mo i a ion
Coas al looding poses an unp eceden ed h ea ha demands ou immedia e a en ion
and ac ion. A ound he wo ld, communi ies, businesses, and i al ecosys ems ace ising
isks as s eng hening seas and ex eme wea he e en s in ensi y. The u gency o his
challenge, accele a ed by clima e change, calls o inno a i e solu ions based on
igo ous scien i ic unde s anding. The esea ch a emp s his c i ical need by
in es iga ing how beaches can p o ec ou coas lines. By expanding ou knowledge o
hese na u al ba ie s and hei esponse o clima e change, we aim o unlock new
possibili ies o coas al p o ec ion. This wo k is no jus abou unde s anding beaches; i
is abou sa egua ding communi ies, p ese ing economies, and p o ec ing he delica e
balance o coas al ecosys ems o u u e gene a ions.
The Eb o Del a se es as bo h a scien i ically op imal case s udy and an a ea o pe sonal
signi icance. G owing up in he egion, my amily's equen excu sions o he Del a
os e ed a deep app ecia ion o his unique coas al en i onmen . These childhood
expe iences o obse ing i s dynamic landscapes, di e se ecosys ems, and cul u al
impo ance ha e de eloped in o a p o essional commi men o unde s anding and
p o ec ing his i al na u al sys em. This pe sonal connec ion p o ides aluable
comp ehension o he Del a's signi icance beyond i s ecological and economic alue. The
Del a's cha ac e is ics o e a equi ing con ex o in es iga ing coas al p o ec ion
s a egies, wi h indings ha hold po en ial applica ions o coas al communi ies
wo ldwide acing simila challenges.
The u gen need o p o ec ou coas lines calls o inno a i e solu ions, and his esea ch
ises o ha challenge. A a ime when coas al communi ies ace clima e h ea s, his
esea ch p o ides i al insigh s o policymake s and enginee s alike. The indings
ca alyse ac ion, empowe ing decision-make s o de elop obus , na u e-based solu ions
ha s eng hen ou coas lines agains clima e change. Th ough his wo k, we a e no
jus s udying coas al esilience, we a e ac i ely shaping a mo e sus ainable and secu e
u u e o ou coas al egions.
16
The Eb o Del a's expe ience o e s aluable insigh s o de eloping adap i e
managemen s a egies in simila coas al en i onmen s wo ldwide, illus a ing he
complex decisions equi ed o p o ec and p ese e ulne able coas al ecosys ems in an
e a o apid en i onmen al change.
The p esen esea ch ocuses on wo dis inc i e coas al a eas wi hin he Eb o Del a ha
exempli y he a ying dynamics and ulne abili ies o his complex ecosys em.
Ma quesa Beach, si ua ed in he no he n hemidel a, and T abucado Beach (Figu e 6)
in he sou he n sec ion se e as ep esen a i e cases o analysing coas al esponse o
clima e change impac s.
Figu e 6: The T abucado Beach: A F agile Poin in he Time-C i ical Ba le a he Eb o Del a. Sou ce: RTVE.
(2023, May 24). Pla ja d he T abucado : la llui a a con a emps pe sal a el Del a de l'Eb e. RTVE.
h ps://www. e.es/ ele ision/20230524/pla ja- abucado -llui a-a-con a emps-del a-
eb e/2446163.sh ml

17
2.2. Coas al looding on beaches: e iew o exis ing equa ions o wa e
un-up and looding on beaches
Wa e un-up quan i ies he maximum e ical ele a ion a wa e achie es abo e he s ill
wa e le el when in e ac ing wi h a coas al slope o s uc u al in e ace. This p ecise
measu emen p o ides comp ehensi e e alua ion o ma ine ene gy in e ac ion wi h
e es ial en i onmen s.
Hence, by cap u ing he e ical displacemen o wa e mo ion, as p esen ed in Figu e 7,
un-up analysis o e s essen ial da a o unde s anding coas al ulne abili y. The
measu emen se es as a key diagnos ic ool o e alua ing he dynamic o ces ha
shape coas al landscape and lood isk.
Figu e 7: De ini ion o Wa e Run-up. Sou ce: Debo ah Villa oel-Lamb (2022). In Quan i ying wa e unup
in da a-spa se loca ions o planning. Jou nal o Coas al Resea ch.
h ps://www. esea chga e.ne /publica ion/359817802_Quan i ying_Wa e_Runup_in_Da a-
Spa se_Loca ions_ o _Planning
Re iewing exis ing equa ions o wa e un-up equi es analysing di e se ma hema ical
o mula ions used o es ima e he heigh ha wa es will each on a beach o a coas al
s uc u e. These equa ions a e c ucial o p edic ing coas al looding and designing
e ec i e coas al de ences. The a ailable o mula ions a y in complexi y and
applicabili y, anging om empi ical o mulas o idealized beaches and s uc u es o
ad anced nume ical models o mo e complex coas al en i onmen s. The choice o
equa ion o model depends on he speci ic condi ions o he si e and he le el o de ail
equi ed o he analysis. Fo accu a e p edic ions, i is c ucial o conside beach slope,
wa e cha ac e is ics, and po en ial idal in luences. They aim o p o ide a
comp ehensi e unde s anding o wa e beha iou and i s impac on coas al a eas.
18
The lowe igu e 8 p esen s a summa y o he mos used un-up equa ions:
Figu e 8: E olu ion o he Wa e Run-up Equa ions. Sou ce: Main.
2.2.1. Hun (1959)
One o he oldes un-up equa ions, Hun 's o mula is based on linea wa e heo y and
is ypically used o s eepe slopes, such as seawalls o e e men s. I ocuses on wa e
un-up on impe meable s uc u es o s eep coas s. I is simple, and pa icula ly e ec i e
o s eepe , impe meable coas al s uc u es. Bes sui ed o : Seawalls, s eep beaches,
ocky sho es. [1]
𝑅=𝛾𝐻𝑠 an(𝜃) (Eq. 1)
Desc ip ion:
- R = un-up heigh .
- γ = empi ical cons an (usually a ound 1.5 o smoo h, impe meable su aces).
- 𝜃 = slope angle.
- Hs = signi ican wa e heigh .
2.2.2. Yamamo o (1972)
This equa ion is o en used in coas al enginee ing o es ima e wa e un-up in a ious
beach condi ions, including hose wi h signi ican wa e heigh s and a ying
wa eleng hs, o en applicable o p elimina y assessmen s. [2]
𝑅=𝜉⋅√𝐻𝑠⋅𝐿 (Eq. 2)
19
Desc ip ion:
- R = un-up heigh .
- Hs = signi ican wa e heigh .
- L = deep wa e wa eleng h.
- 𝜉 = I iba en numbe (dimensionless pa ame e ).
2.2.3. Holman (1986)
This is a simple equa ion ha p edic s he maximum un-up heigh based on he wa e
s eepness and beach slope. Bes sui ed o : Coas al si es wi h mild slopes and sandy
beaches. [3]
𝑅2% =𝛼𝛽𝑓√𝐻𝑠𝐿0 (Eq. 3)
Desc ip ion:
- R2% is he maximum un-up heigh .
- 𝛼 is an empi ical cons an ha depends on he beach slope.
- β = o esho e slope.
- Hs = signi ican wa e heigh .
- L0 = deep wa e wa eleng h.
2.2.4. Kobayashi and Tamu a (1986)
Rela es signi ican wa e heigh o wa eleng h o es ima ing wa e un-up. P ima ily
used o sandy beaches, his equa ion is use ul when dealing wi h un-up unde
condi ions o a ying wa e heigh and leng h. I is pa icula ly applicable in coas al a eas
wi h consis en wa e pa e ns. [4]
𝑅2% =𝐶⋅𝐻𝑠⋅(𝐿
𝐻𝑠)𝑛 (Eq. 4)
Desc ip ion:
- R2% = un-up heigh .
- Hs = signi ican wa e heigh .
- L = deep wa e wa eleng h.
- C = an empi ical cons an , which depends on beach slope and wa e condi ions.
- n= is ano he empi ical exponen ha accoun s o he e ec s o he beach slope
and wa e b eaking cha ac e is ics.
20
2.2.5. Mase (1989)
Mase de eloped an empi ical o mula based on ield da a om na u al beaches. I is
o en used in coas al haza d assessmen s. Uses ield da a o alida e p edic ions. Bes
sui ed o : Na u al beaches, mild o mode a e slopes. [5]
𝑅
=0,56𝐻𝑠(𝐿0
𝐻𝑠)2𝛽𝑓
0,5
(Eq. 5)
Desc ip ion:
- R = un-up heigh .
- Hs = signi ican wa e heigh .
- L0 = deep wa e wa eleng h.
- β = beach slope.
2.2.6. Nielsen & Hanslow (1991)
This equa ion was designed o sandy beaches and ocuses on s o m wa e un-up. I is
commonly used o beach e osion s udies du ing high-ene gy wa e condi ions. Simple
and use ul o s o m condi ions. Bes sui ed o : Sandy, e odible coas lines du ing
s o ms. [6]
𝑅
2%=2,3⋅𝐻𝑠 (Eq. 6)
Desc ip ion:
- R = un-up heigh .
- Hs = signi ican wa e heigh .
2.2.7. Van de Mee & S am (1992)
This o mula is speci ically designed o ocky slopes and ubble-mound b eakwa e s. I
inco po a es he oughness and pe meabili y o he s uc u e, making i sui able o
enginee ed coas al de ences. Tailo ed o ough and pe meable s uc u es, such as
b eakwa e s. Bes sui ed o : Coas al de ences like b eakwa e s and e e men s. [7]
𝑅
2%=𝜃𝛾𝑠𝐻𝑠𝜀𝑚 (Eq. 7)
21
Desc ip ion:
- R2% = 2% exceedance un-up heigh .
- 𝜃 = slope angle.
- γs = educ ion ac o (accoun s o oughness and pe meabili y).
- Hs = signi ican wa e heigh .
- ξm = su simila i y pa ame e .
2.2.8. S ockdon e al. (2006)
This model includes bo h swash and in ag a i y wa e con ibu ions. Use ul o
p edic ing ex eme un-up. Bes sui ed o : Sandy beaches, low-g adien sho es. [8]
𝑅2% =1,1(0,35𝛽𝑓√𝐻𝑠𝐿0+(𝐻𝑠𝐿0(0,563𝛽𝑓2+0,004))1/2
2) (Eq. 8)
Desc ip ion:
- R2% = he 2% exceedance un-up heigh .
- β = he o esho e beach slope.
- Hs = he signi ican wa e heigh a b eaking.
- L0=gTp
2
2π is he deep wa e wa eleng h (g is g a i a ional accele a ion, and Tp is
he wa e peak pe iod), calcula ed om Hs.
2.2.9. Flick and Mu phey (2008)
This model is pa icula ly use ul o a ying beach p o iles whe e slope and sedimen
size ha e signi ican in luences on wa e un-up. I is ele an in en i onmen s whe e
complex in e ac ions be ween wa e dynamics and beach mo phology occu , use ul such
as in design and sa e y assessmen s. [9]
𝑅 𝑚𝑎𝑥 =1,1⋅ 𝐻𝑠
√2⋅𝑐𝑜𝑠𝜃 (Eq. 9)
Desc ip ion:
- Rmax = un-up heigh .
- Hs = signi ican wa e heigh .
- θ = slope angle.

22
2.2.10. Mase and Shibayama (2009)
This equa ion is pa icula ly use ul o sandy beaches and is ocused on empi ical
ela ionships, making i ele an in en i onmen s whe e he wa e heigh and
wa eleng h a e p ima y ac o s. So applied in coas al managemen scena ios whe e
empi ical da a is a ailable o calib a ion, is also use ul in designing coas al de ences.
[10]
𝑅2% =1,5⋅𝐻𝑠⋅(𝐿
𝐻𝑠)0,5 (Eq. 10)
Desc ip ion:
- R2% = he 2% exceedance un-up heigh .
- Hs = signi ican wa e heigh .
2.2.11. Summa y o wa e un-up equa ions
The de elopmen o wa e un-up equa ions o e ime e lec s a p og ession owa ds
mo e comp ehensi e and accu a e p edic ions o coas al looding. This ch onological
e olu ion demons a es how inco po a ing a b oade ange o a iables, e ining
empi ical ela ionships, and ad ances in he heo e ical unde s anding o wa e-coas al
in e ac ions ha e signi ican ly enhanced he p ecision o hese models.
Ea ly equa ions, such as Hun (1959), ocused p ima ily on key ac o s like wa e heigh
and beach slope. As he ield ad anced, subsequen equa ions began o inco po a e
addi ional pa ame e s such as wa eleng h, wa e pe iod, and sedimen cha ac e is ics.
Newe equa ions, including Mase (1989), Flick and Mu phey (2008), and Mase and
Shibayama (2009), u he e ined p edic ions by accoun ing o a ying beach p o iles,
slope angles, and empi ical ela ionships speci ic o di e en coas al en i onmen s.
This p og ession unde sco es he inc easing complexi y o hese models, wi h each
imp o emen ep esen ing a deepe unde s anding o he dynamic in e ac ions
be ween wa es and coas al sys ems. Including mo e a iables and imp o ing empi ical
models ha e led o equa ions be e sui ed o speci ic coas al se ings, such as sandy
beaches, ocky slopes, o low-g adien sho es.
In conclusion, he e olu ion o wa e un-up equa ions has esul ed in a se o ools ha
enable mo e e ec i e coas al managemen and mi iga ion s a egies.
Howe e , he choice o equa ion depends on se e al ac o s, including he speci ic
coas al en i onmen , da a a ailabili y, and he le el o complexi y equi ed o he
analysis. Some equa ions, like Holman (1986), a e simple and equi e ewe inpu s,
while o he s, such as S ockdon e al. (2006), a e mo e complex and need a wide ange
o da a.
23
2.3. Selec ing he App op ia e Wa e Run-up Equa ion o he Eb o Del a
When s udying wa e un-up o a speci ic coas al a ea like he Eb o Del a, i is c ucial o
choose an equa ion ha aligns wi h he egion's mo phological and physical
cha ac e is ics. Eb o Del a ea u es a low-lying, gen ly sloping p o ile wi h sandy beaches
and we lands, indica ing ha equa ions ailo ed o low-g adien , sedimen a y
en i onmen s a e mos sui able.
While simple equa ions like Holman (1986) can p o ide quick es ima es o maximum
un-up heigh based on wa e s eepness and beach slope, he S ockdon e al. (2006)
equa ion is gene ally mo e app op ia e o en i onmen s like he Eb o Del a.
The S ockdon e al. (2006) [8] equa ion o e s se e al ad an ages ha make i he
p e e ed choice o he coas line being s udied:
1. Comp ehensi e App oach, as he equa ion includes bo h swash and in ag a i y
e ms, p o iding a mo e accu a e ep esen a ion o he p ocesses occu ing on
low-g adien , sedimen a y coas lines.
2. Pa ame e iza ion, as he equa ion is pa ame e ized wi h coe icien s ha adap o
di e en beach p o iles and wa e condi ions, allowing o b oade applicabili y
ac oss a ious coas al en i onmen s.
3. Adap abili y o Beach Condi ions, as he S ockdon equa ion adap s o di e en
coas al condi ions by using ei he he ull o simpli ied e sion based on he
I iba en numbe (ξ). The I iba en numbe is a dimensionless pa ame e ha
ep esen s he ype o beach, wi h low alues (ξ < 3) indica ing dissipa i e
condi ions and high alues (ξ ≥ 3) indica ing e lec i e condi ions, o dissipa i e
beaches like he Eb o Del a, he ull equa ion is used, while o e lec i e beaches,
he simpli ied equa ion neglec ing he in ag a i y e m is applied.
4. Sui abili y o Dissipa i e Condi ions, as he Eb o Del a is cha ac e ized by
ex emely dissipa i e condi ions due o i s gen le slope, wide su zone, ine
sedimen s, and dominance o low- equency wa es. The S ockdon equa ion,
pa icula ly he ull e sion, is e ec i e in cap u ing un-up dynamics in such
dissipa i e en i onmen s.
5. Accu acy: Al hough all empi ical equa ions ha e associa ed unce ain ies, he
S ockdon equa ion has a ela i ely low e o o app oxima ely ±0.32 me es o he
2% exceedance un-up heigh p edic ion when using he ull equa ion o
dissipa i e condi ions. [8]
This makes i a eliable choice o he Eb o Del a gi en he low-g adien , sedimen a y
na u e o he Eb o Del a’s coas line and he equa ion's s eng hs in ep esen ing swash
and in ag a i y p ocesses, he S ockdon e al. (2006) equa ion is he mos app op ia e
choice o s udying wa e un-up in his egion.
24
2.3.1. The I iba en numbe
Howe e , his exp ession has a a ia ion o ex emely dissipa i e condi ions, due o o
e lec i e beaches, he equa ion o un-up a 2% exceedance ele a ions can be
simpli ied by neglec ing he in ag a i y con ibu ion ( he 0.004 e m). Hence, The
S ockdon e al. (2006) equa ion adap s o di e en coas al condi ions by using ei he he
ull (Eq. 8) o simpli ied (Eq. 12) e sion based on he I iba en numbe (ξ) (Eq. 11).
The I iba en numbe is a dimensionless pa ame e ha ep esen s he ype o beach,
wi h low alues indica ing dissipa i e condi ions and high alues indica ing e lec i e
condi ions.
The I iba en numbe is calcula ed as ollows (Eq. 11):
ξ = 𝛽𝑓
√𝐻𝑠
𝐿0
(Eq. 11)
Whe e:
- Β is he beach slope.
- Hs is he signi ican wa e heigh a b eaking.
- L0=gTp
2
2π is he deep wa e wa eleng h (g is g a i a ional accele a ion, and T is
he wa e peak pe iod), calcula ed om Hs.
Equa ions and associa ed e o s based on he I iba en numbe :
1. Fo dissipa i e condi ions (ξ < 3):
Full equa ion:
𝑅2% =1,1(0,35𝛽𝑓√𝐻𝑠𝐿0+(𝐻𝑠𝐿0(0,563𝛽𝑓2+0,004))1/2
2) (Eq. 8)
- E o : ± 0.32 me es [8].
Whe e:
- R2% is he 2% exceedance un-up heigh ( he le el exceeded by 2% o he wa es).
- β is he o esho e beach slope.
- Hs is he signi ican wa e heigh a b eaking.
- L0 is he deep wa e wa eleng h
25
2. Fo e lec i e condi ions (ξ ≥ 3):
Simpli ied equa ion:
𝑅2% =0,73𝛽𝑓√𝐻𝑠𝐿0 (Eq. 12)
- E o : ± 0.47 me es [8].
I is impo an o no e ha he e o could a y sligh ly depending on local condi ions,
bu his p o ides a gene al idea o he accu acy. I used in en i onmen s like he Eb o
Del a, whe e he coas al p o ile migh no di e d as ically om he da a se S ockdon
equa ion (gen ly sloping sandy beaches), he e o should be wi hin a simila ange.
32
3. Code Me hodology
3.1. In oduc ion
A eliabili y analysis o beach de ence sys ems agains s o m impac s has been
de eloped in MATLAB, Appendix 1 con ains he comple e code used, while Appendix 2
p o ides an o e iew o he unc ions u ilized wi hin he code.
This enginee ing ool e alua es beach pe o mance unde p esen and u u e clima e
scena ios h ough cus omizable pa ame e s. The code implemen s local cha ac e is ics
such as beach slope, c i ical wa e le el heigh , alid wa e di ec ions, and subsidence
a es o assess si es' eliabili y and de elop e idence-based adap a ion s a egies.
The p obabilis ic me hodology ensu es obus eliabili y es ima es h ough a Mon e
Ca lo app oach wi h a high numbe o epe i ions, accoun ing o unce ain ies in bo h
clima e p ojec ions (RCP4.5 and RCP8.5 scena ios), subsidence a es, and en i onmen al
condi ions (as onomical ide, me eo ological esidue, and wa e pa ame e s) om 2025
o 2100. The amewo k's modula s uc u e enables he e alua ion o indi idual
beaches wi hin del a egions by adjus ing si e-speci ic pa ame e s.
3.1.1. Mon e Ca lo Simula ion Model o Coas al Flooding Risk Assessmen
The analysis employs a Mon e Ca lo simula ion model wi h 1000000 (1M) i e a ions
eaching obus con e gence o assess coas al looding isks. Th ough andom sampling
and s a is ical modelling, he model gene a es comp ehensi e es ima ions o looding
scena ios, p o iding essen ial insigh s o coas al managemen .
The me hod inco po a es unce ain y by assigning p obabili y dis ibu ions o inpu
pa ame e s, based on his o ical da a, and s a is ical analysis. The model e alua es
looding isks and in as uc u e impac s while accu a ely ep esen ing pa ame e
in e dependencies. This p ocess a ends o he na u al complexi y o coas al sys ems,
pa icula ly he andom beha iou o hyd odynamic and mo phological pa ame e s [28].
The comple e sampling ensu es he analysis cap u es he ull ange o possible scena ios
while accoun ing o ela ionships be ween coas al s a e indica o s and nume ical
models. This conside able numbe o i e a ions enables an accu a e ep esen a ion o
s o m e en s du ing ex eme condi ions, pa icula ly c ucial o coas al looding analysis.
The model ensu es con e gence h ough p og essi e analysis o inc easing sample sizes,
p o iding con idence in he s abili y o he esul s.

33
3.1.2. As onomical Tide Modelling
The as onomical ide modelling uses da a om Pue os del Es ado (PORTUS) [39]
oceanog aphic da abase (Table 2), speci ically om he Ta agona ide gauge s a ion.
The idal analysis e ealed 22 signi ican cons i uen s, implemen ed h ough he
ollowing equa ion (Eq. 15):
𝜂(𝑡) = 𝑍₀ + 𝛴(𝐴ᵢ 𝑐𝑜𝑠(𝜔ᵢ𝑡 + 𝜑ᵢ)) (Eq. 15)
Whe e:
- η( ): wa e le el a ime .
- Z₀: mean sea le el (33.37 cm om Ta agona ide gauge)[39] .
- Aᵢ: ampli ude o cons i uen .
- ωᵢ: angula equency.
- φᵢ : phase angle.
Table 2: Ha monics, Tide Guge o Ta agona. Sou ce: Pue os del Es ado. h ps://po us.es/
34
Tidal ha monics, including luna (e.g., M2, N2), sola (e.g., S2), and o he ypes (e.g., K1,
O1), ep esen he g a i a ional in luences o he moon, sun, and Ea h’s dynamics on
idal pa e ns. Luna and sola ha monics d i e semi-diu nal ides wi h di e ing
equencies, while o he s cap u e diu nal cycles and complex in e ac ions. Each
ha monic is cha ac e ized by i s equency (cycles pe hou ), ampli ude (heigh
con ibu ion), and phase ( ime o se ela i e o a e e ence poin ), enabling p ecise ide
p edic ions.
The 22 ha monic cons i uen s a e in eg a ed in o he Mon e Ca lo simula ion o
gene a e ealis ic idal a ia ions, which es ablish a con inuous idal unc ion h oughou
he yea . By gene a ing andom hou s wi hin he 8760 hou s o a non-leap yea , speci ic
as onomical ide le els a e ob ained o each Mon e Ca lo i e a ion. This andomiza ion
ensu es he analysis cap u es idal a ia ions ac oss all empo al scales, om daily o
seasonal cycles.
Fi s ly, he gene al cha ac e is ics o he model a e de ined o ob ain he necessa y
pa ame e s o an accu a e analysis.
Code 1: Gene al Cha ac e is ics
As i is de e mined in he Code 1 ex sec ion, a andom hou o e a yea is gene a ed
o assign de e mined condi ions o he ide le el compu a ion. This me hod p o ides
ealis ic as onomical ide a ia ions o eliabili y analysis while main aining he
s a is ical independence o samples.
Then, once he andom hou o he yea is ob ained, applying he s a ed p ocedu e,
he ollowing Code 2 was ob ained:
- Random hou gene a ion: h ∈ [1, 8760].
- Tide le el calcula ion: Equa ion 15
- Con e sion o me es: esul s/100.
%% Gene al ca ac e is ics
o al_hou s = 8760; % To al hou s in a non-leap yea
num_ epe i ions = 1000000; % Numbe o epe i ions
g = 9.81; % G a i a ional accele a ion (m/s^2)
% Gene a ion o consis en andom hou selec ions o bo h analyses
andom_hou s = andi( o al_hou s, num_ epe i ions, 1);
35
%% 2. Calcula e As onomical Tide
Z0 = 33.37; % Mean sea le el in cm
% Ha monic componen s
ha monics = {
'SA', 0.000114, 7.62, 249.24;
'M2', 0.080511, 3.97, 207.67;
'K1', 0.041781, 3.7, 164.87;
'O1', 0.038731, 2.4, 103.02;
'S2', 0.083333, 1.35, 229.34;
'P1', 0.041553, 1.26, 160.36;
'N2', 0.078999, 0.86, 196.71;
'S1', 0.041667, 0.66, 261.22;
'M4', 0.161023, 0.5, 346.59;
'K2', 0.083561, 0.4, 223.79;
'Q1', 0.037219, 0.31, 53.58;
'MS4', 0.163845, 0.31, 51.29;
'MN4', 0.159511, 0.19, 304.58;
'NU2', 0.079202, 0.15, 199.07;
'M3', 0.120767, 0.15, 157.55;
'2N2', 0.077487, 0.14, 184.27;
'MU2', 0.077689, 0.14, 173.01;
'L2', 0.082024, 0.11, 216.42;
'T2', 0.083219, 0.1, 193.49;
'MK4', 0.164073, 0.09, 55.84;
'SK3', 0.125114, 0.08, 109.04;
'SN4', 0.162333, 0.05, 6.84;
};
Code 2: As onomical Tide Compu a ion
equencies = cell2ma (ha monics(:, 2));
ampli udes = cell2ma (ha monics(:, 3));
phases_ adians = deg2 ad(cell2ma (ha monics(:, 4)));
as onomical_ ide = Z0 + sum(ampli udes .* cos(2 * pi * equencies .*
andom_hou s' + phases_ adians), 1)';
as onomical_ ide = as onomical_ ide / 100; % Con e o me es
3.1.3. Me eo ological esidue
The me eo ological esidue is a c i ical componen in coas al lood isk analysis,
pa icula ly when assessing s o m e en s. I accoun s o wa e le el a ia ions caused
by h ee key ac o s:
1. A mosphe ic p essu e changes: Low-p essu e sys ems du ing s o ms cause
wa e le els o ise due o he in e se ba ome e e ec .
2. Wind s ess e ec s: S ong winds accompanying s o ms can pile up wa e
agains he coas line, leading o ele a ed wa e le els and inc eased lood isk.
3. La ge-scale oceanog aphic phenomena: S o ms can in e ac wi h ides and
ocean cu en s, u he in luencing wa e le els. Fo example, a s o m coinciding
wi h a high ide can esul in conside ably highe wa e le els.
36
Including he me eo ological esidual in coas al lood isk analysis ensu es a mo e
accu a e assessmen o po en ial s o m su ges and associa ed lood isks. I p o ides a
ealis ic ep esen a ion o he wa e le el a ia ions ha can occu du ing s o m e en s.
Neglec ing he me eo ological esidual can lead o an unde es ima ion o lood isk,
esul ing in inadequa e coas al p o ec ion measu es, hence, inco po a ing his
componen is essen ial o e ec i e coas al managemen o mi iga e he impac s o
coas al looding.
To model he me eo ological esidual o he Po o Ta agona, a Gaussian p obabili y
dis ibu ion is employed, wi h alues anging om 0.3 o 0.7. This in e al is de i ed
om empi ical da a p esen ed in he Summa y o Pa ame e s Rela ed o Sea Le el and
Tide ha A ec Po Design and Ope a ion Condi ions speci ic o he po [39].
The Gaussian dis ibu ion is cha ac e ized by wo key pa ame e s: he mean (μ) and he
s anda d de ia ion (σ). In his model, he mean is se o 0.5, ep esen ing he cen e o
he dis ibu ion, while a educed s anda d de ia ion o 0.06 ensu es ha he gene a ed
alues all wi hin he desi ed ange. To achie e his, a unca ed no mal dis ibu ion
app oach is applied, gene a ing andom alues om a s anda d no mal dis ibu ion and
ans o ming hem o i he speci ied pa ame e s. The esul ing alues a e il e ed o
e ain only hose wi hin he de ined bounds o 0.3 o 0.7.
This app oach ensu es ha he gene a ed me eo ological esidual alues a e bo h
ealis ic and ep esen a i e o he condi ions a he Po o Ta agona. These alues can
be u ilized in coas al lood isk analysis o accoun o po en ial wa e le el a ia ions
caused by a mosphe ic p essu e changes, wind s ess e ec s, and la ge-scale
oceanog aphic phenomena du ing s o m e en s.
By i ing he Gaussian p obabili y dis ibu ion o empi ical da a, he model p o ides a
obus and eliable me hod o cap u ing he inhe en a iabili y and unce ain y o
me eo ological esiduals, ensu ing he gene a ed alues align wi h he speci ic
cha ac e is ics o he Po o Ta agona.
Following he ou lined me hodology, he Code 3 was de i ed:
Code 3: Gene a ion o he Me eo ological Residue
%% 3. Gene a e Me eo ological Residue
mu = 0.5; % Mean (cen e )
sigma = 0.06; % Reduced s anda d de ia ion o be e sp ead wi hin bounds
% Gene a e using unca ed no mal dis ibu ion app oach
z = andn(num_ epe i ions*2, 1); % Gene a e ex a alues o unca ion
z = mu + sigma * z;
alid_idx = z >= 0.3 & z <= 0.7; % Find alues wi hin bounds
esidue_ alues = z( alid_idx); % Keep only alid alues
esidue_ alues = esidue_ alues(1:num_ epe i ions); % Take equi ed numbe
o samples
37
3.1.4. Wa e Heigh Analysis and Weibull Dis ibu ion
The signi ican wa e heigh (Hs) a he buoy o Ta agona [39] is modelled using a
Weibull p obabili y dis ibu ion, which e ec i ely ep esen s bo h ypical wa e heigh s
and ex eme alues. The model employs a wo- egime app oach o cap u e wa e heigh
cha ac e is ics unde mode a e and ex eme clima e condi ions.
Mode a e Clima e Regime: Fo he mode a e clima e egime, he Weibull in e se
unc ion desc ibing he signi ican heigh o he wa es in deep wa e is used (Eq. 16):
𝐻𝑠 = 𝐵 + 𝐴(−𝑙𝑛(1−𝑃))(1/𝐶) (Eq. 16)
The pa ame e s a e as ollows:
- A, being he scale pa ame e .
- B, being he loca ion pa ame e .
- C, being he shape pa ame e .
Seasonal adjus men s a e made o he pa ame e s o e lec a ying wa e condi ions
h oughou he yea :
- Win e : A=0.79, B=0.22, C=1.04
- Sp ing: A=1.04, B=-0.06, C=1.57
- Summe : A=0.4, B=0.14, C=1.15
- Au umn: A=1.03, B=-0.01, C=1.56
The model gene a es andom hou s, he same o hose used in de e mining
as onomical ides, and andom p obabili y alues be ween 0 and 1. These p obabili y
alues a e used as inpu o he Weibull in e se unc ion, which assigns cons an s based
on he espec i e season. I hen calcula es he wa e heigh (Hs) using he Weibull
in e se unc ion.
Ex eme Value Regime (Hs ≥ 3m): I he compu ed Hs unde he mode a e clima e
egime is equal o o g ea e han 3 me es, which is he h eshold alue desc ibed in
he Ta agona’s buoy o ex eme e en s [39], he model eassigns a p obabili y o his
alue and ansi ions o he ex eme alue egime. This ansi ion in ol es applying
adjus ed cons an s o ex eme condi ions using a modi ied Weibull dis ibu ion.

38
Only he ecalcula ed wa e heigh s a e conside ed alid, as he analysis ocuses on
s o m e en s, which a e he mos c i ical and can de e mine he exceedance o he c es
le el. In his egime, a modi ied Weibull dis ibu ion is used o calcula e he wa e heigh
(Eq. 16):
𝐻𝑠 = 𝛼+ 𝛽(−𝑙𝑛(1−𝑃))(1/𝛾) (Eq. 16)
The pa ame e s o he ex eme alue egime a e:
- α = 3.1
- β = 0.44
- γ = 0.76
The ex eme alue egime ep esen s s o m e en s, which a e o pa icula conce n o
coas al and po managemen . I cap u es he wa e heigh s ha exceed he signi ican
heigh limi wi hin he a e age egime.
This wa e heigh analysis is based on empi ical da a eco ded by he Ta agona Buoy,
ensu ing an accu a e ep esen a ion o he local wa e clima e speci ic o he Po o
Ta agona. The p ima y pa ame e s we e de i ed om he "A e age Wa e Clima e"
da ase , while da a o s o m e en s was sou ced om he documen i led "Maximum
Wa e Ex emes by Di ec ions (Signi ican Wa e Heigh )”. [39]
39
Then, because o he speci ied me hodology, Code sec ion 4 was de eloped, as ollows:
Code 4: Gene a ion o Signi ican Wa e Heigh s unde di e en clima e egimes.
%% 4. Calcula e Wa e Heigh s and Seasonal Pa ame e s
% De ine mon h anges
mon h_ anges = [
1, 744; % Janua y (31 days)
745, 1416; % Feb ua y (28 days)
1417, 2160; % Ma ch (31 days)
2161, 2880; % Ap il (30 days)
2881, 3624; % May (31 days)
3625, 4344; % June (30 days)
4345, 5088; % July (31 days)
5089, 5832; % Augus (31 days)
5833, 6552; % Sep embe (30 days)
6553, 7296; % Oc obe (31 days)
7297, 8016; % No embe (30 days)
8017, 8760 % Decembe (31 days)
];
% Ini ialize a ays
mon hs = ze os(num_ epe i ions, 1);
A = ze os(num_ epe i ions, 1);
B = ze os(num_ epe i ions, 1);
C = ze os(num_ epe i ions, 1);
% Assign seasonal cons an s
o i = 1:num_ epe i ions
hou = andom_hou s(i);
o mon h = 1:12
i hou >= mon h_ anges(mon h, 1) && hou <=
mon h_ anges(mon h, 2)
mon hs(i) = mon h;
% Assign cons an s based on season
swi ch ue
case ismembe (mon h, [12, 1, 2]) % Win e
[A(i), B(i), C(i)] = deal(0.79, 0.22, 1.04);
case ismembe (mon h, 3:5) % Sp ing
[A(i), B(i), C(i)] = deal(1.04, -0.06, 1.57);
case ismembe (mon h, 6:8) % Summe
[A(i), B(i), C(i)] = deal(0.4, 0.14, 1.15);
case ismembe (mon h, 9:11) % Au umn
[A(i), B(i), C(i)] = deal(1.03, -0.01, 1.56);
end
b eak;
end
end
end
% Calcula e ini ial and adjus ed Hs
P = and(num_ epe i ions, 1);
Hs_ini ial = B + A .* (-log(1 - P)).^(1 ./ C);
% Pa ame e s o Hs adjus men
be a = 0.44;
alpha = 3.1;
gamma = 0.76;
adjus _indices = Hs_ini ial >= 3;
40
3.1.5. Di ec ional Wa e Analysis unde S o m Condi ions
The di ec ional wa e analysis is a c ucial componen o he wa e heigh model, ocusing
on assigning wa e di ec ions o he gene a ed signi ican wa e heigh s (Hs). The analysis
employs a disc e e p obabili y dis ibu ion conside ing eigh ca dinal and in e ca dinal
di ec ions (N, NE, E, SE, S, SW, W, NW) o ep esen he p ima y di ec ional pa e ns.
The model assigns p obabili ies o each di ec ion based on he obse ed di ec ional
dis ibu ion ob ained om his o ical ex eme e en s da a collec ed by he buoy o
Ta agona [39]. This da a (Figu e 11) p o ides help ul pe cep ions o he di ec ional
cha ac e is ics o ex eme wa e condi ions, allowing o a mo e accu a e ep esen a ion
o wo s -case scena ios.
To u he e ine he di ec ion assignmen , he model inco po a es si e-speci ic alid
app oach di ec ions, accoun ing o local ac o s such as ba hyme y and coas line
o ien a ion. Wa es wi h di ec ions ou side he alid anges a e il e ed ou o ensu e
only ealis ic and physically possible di ec ions a e conside ed. I is impo an o no e
ha he alid di ec ions o incidence a e de ined as a iables speci ic o each beach, and
only he Hs alues ha ollow hese di ec ions will be e ained o u he analysis.
The assignmen o wa e di ec ions o he gene a ed Hs alues, in deep wa e unde
s o m e en s, is pe o med using he cumula i e p obabili y me hod. The p obabili ies
o each di ec ion a e con e ed in o cumula i e p obabili ies, ep esen ing he
likelihood o a wa e coming om ha di ec ion o any p eceding di ec ion. Fo each
il e ed Hs alue, a andom numbe be ween 0 and 1 is gene a ed and ma ched wi h he
co esponding di ec ion based on he cumula i e p obabili y dis ibu ion.
% Ini ialize and calcula e adjus ed Hs
Hs_adjus ed = nan(num_ epe i ions, 1);
P_adjus ed = and(sum(adjus _indices), 1);
Hs_adjus ed(adjus _indices) = alpha + be a .* (-log(1 -
P_adjus ed)).^(1 ./ gamma);
41
Figu e 11: Wa e Incidence Di ec ion P obabili y. Sou ce: Pue os del Es ado, Ta agona Bouy.
h ps://po us.es/
Hence, employing he desc ibed me hodology, Code 5 his sec ion was ex ac ed:
Code 5: Di ec ion assigna ion o ex eme alues.
%% 5. Wa e Di ec ion Analysis
wa e_di ec ions = {'N', 'NE', 'E', 'SE', 'S', 'SW', 'W', 'NW'};
di ec ion_p obabili ies = [4, 5, 25, 15, 24, 9, 9, 9];
% Calcula e cumula i e p obabili ies
cumula i e_p obs = cumsum(di ec ion_p obabili ies) /
sum(di ec ion_p obabili ies);
p ob_edges = [0; cumula i e_p obs(:)];
% Gene a e and assign di ec ions
P_di ec ion = and(sum(adjus _indices), 1);
di ec ion_indices = disc e ize(P_di ec ion, p ob_edges);
% Con e indices o di ec ions
wa e_di ec ions_a ay = cell(num_ epe i ions, 1);
wa e_di ec ions_a ay(adjus _indices) =
wa e_di ec ions(di ec ion_indices);
% Fil e alid di ec ions
wa e_di s_ca = ca ego ical(wa e_di ec ions_a ay(adjus _indices));
alid_di s_ca = ca ego ical( alid_di ec ions);
alid_di ec ion_mask = ismembe (wa e_di s_ca , alid_di s_ca );
alid_di ec ion_indices = alse(num_ epe i ions, 1);
alid_di ec ion_indices(adjus _indices) = alid_di ec ion_mask;
% C ea e inal Hs
Hs_ inal = nan(num_ epe i ions, 1);
Hs_ inal( alid_di ec ion_indices) = Hs_adjus ed( alid_di ec ion_indices);
48
3.1.9. Con e gence Analysis o Mon e Ca lo Simula ion Resul s
Mon e Ca lo simula ion, while powe ul o analysing coas al looding isks, equi es
ca e ul alida ion o ensu e i s esul s a e eliable and s able. The con e gence analysis
se es as a c i ical ool o e i ying ha he numbe o simula ions pe o med is
su icien o p oduce dependable p obabili y es ima es.
In his s udy, he con e gence analysis examines how he calcula ed p obabili y o
ailu e s abilizes as he numbe o samples inc eases, which means ha he inal
p obabili y o ailu e conside s all he p e ious i e a ions, so he la alue ob ained o
his p obabili y o ailu e will be he mos p ecise.
The implemen a ion ollows a s uc u ed p ocess, analysing da a in inc emen s o 50
samples pe s ep, wi h a o al limi o 1M i e a ions. To ensu e he consis ency o he
esul s, a mo ing window o 200 samples is used o con inuously e alua e s abili y o e
ime [27].
The analysis employs a ole ance h eshold o 0.0001 (0.01%) o de e mining
con e gence. This means ha when he s anda d de ia ion o ailu e p obabili ies wi hin
he mo ing window alls below his h eshold conce ning he inal p obabili y o ailu e,
i is conside ed he solu ion o ha e con e ged. This s ic ole ance ensu es high
con idence in he inal p obabili y es ima es.
Fo each c i ical heigh , clima e scena io (Cu en , RCP4.5, and RCP8.5) and yea o
s udy, he analysis acks how he p obabili y o ailu e e ol es wi h inc easing sample
size. Con e gence plo s isualize his e olu ion, showing bo h he p og ession owa d
s able alues and he poin a which con e gence is achie ed.
The con e gence analysis e eals se e al impo an aspec s:
- Fi s , i iden i ies he minimum numbe o simula ions equi ed o achie e
eliable esul s, helping o op imize compu a ional esou ces in u u e
s udies while ensu ing accu acy.
- Second, i assesses he s abili y o p obabili y es ima es ac oss di e en
scena ios. Analysing he con e gence beha iou helps de e mine whe he
ce ain scena ios equi e addi ional simula ions o achie e consis en esul s,
indica ing a highe le el o a iabili y in hose condi ions.
3.1.10. Reliabili y Analysis o S o m-Induced Coas al Flooding
Coas al in as uc u e aces i s g ea es challenges du ing s o m e en s, which can
gene a e ex eme wa e le els and po en ially ca as ophic looding. While ypical sea
condi ions a ely h ea en coas al s uc u es, s o m e en s can p oduce combina ions
o high wa es, s o m su ges, and ele a ed wa e le els ha pose signi ican isks.
The e o e, eliabili y analysis in coas al enginee ing ocuses p ima ily on hese ex eme
condi ions, as hey ep esen he scena ios mos likely o cause damage o ailu e.

49
The eliabili y index (β) quan i ies a coas al sys em's sa e y le el agains looding du ing
s o m e en s. This dimensionless pa ame e measu es he numbe o s anda d
de ia ions be ween no mal ope a ing condi ions and po en ial ailu e poin s, exp essed
as (Eq. 20) [34]:
𝛽 = 𝐸[𝑀𝑆] / 𝜎[𝑀𝑆] (Eq. 20)
Whe e:
- E[MS] ep esen s he expec ed alue o he ma gin o sa e y.
- σ[MS] deno es i s s anda d de ia ion.
- The ma gin o sa e y (MS) (Eq. 21) is calcula ed as he di e ence be ween
he c i ical heigh (Hc) and he o al wa e le el (TWL) (Eq. 13) du ing s o m
condi ions:
𝑀𝑆 = 𝐻𝑐 – 𝑇𝑊𝐿 (Eq. 21)
Fo coas al looding analysis, acco ding o da a p o ided in Table 3, he eliabili y index
in e p e a ion ocuses speci ically on s o m e en pe o mance. Values abo e 3.0
indica e obus p o ec ion agains se e e s o ms, while alues below 2.0 sugges
ulne abili y o s o m-induced looding. The minimum ecommended h eshold o
coas al p o ec ion sys ems is β = 2.5, ep esen ing a balance be ween sa e y and
economic easibili y [29].
Table 3: Reliabili y Index Range In e p e a ion. Sou ce: CUR/TAW. (1990). P obabilis ic design o lood
de ences (Repo No. 141). Technical Ad iso y Commi ee on Wa e De ences, Cen e o Ci il Enginee ing
Resea ch and Codes.
Reliabili y Index (β)
Range
In e p e a ion o Coas al Flooding P o ec ion Du ing S o m E en s
β > 3
Robus p o ec ion agains se e e s o ms.
2.0 ≤ β ≤ 3.0
Mode a e p o ec ion, mee s minimum ecommended h eshold o
β = 2.5.
β < 2
Vulne abili y o s o m-induced looding.
Gi en he speci ic na u e o s o m-induced looding, he eliabili y analysis employs
a ge ed il e ing o ocus on po en ially haza dous condi ions. This il e ing p ocess
conside s wo c ucial aspec s o s o m e en s.
The analysis ocuses on wa e di ec ions and heigh s ha a e mos ele an o s o m
condi ions a he si e. Only wa e di ec ions ha align wi h ypical s o m pa e ns and
he si e's geog aphical exposu e a e conside ed, ensu ing ha he eliabili y assessmen
e lec s ac ual s o m isks. Addi ionally, he analysis includes only wa e heigh s
cha ac e is ic o s o m e en s, concen a ing on condi ions capable o causing
signi ican looding.
50
By excluding mino wa e s a es ha do no pose a meaning ul h ea o coas al
in as uc u e, he eliabili y index p o ides a mo e accu a e ep esen a ion o he
sys em's pe o mance unde uly haza dous condi ions ha could lead o in as uc u e
damage o looding., a he han being dilu ed by non- h ea ening condi ions.
The me hodology awa ds ha while coas al sys ems may demons a e high eliabili y
unde no mal condi ions, hei pe o mance du ing s o ms de e mines hei p o ec i e
alue, ensu ing sa e y emain e ec i e agains he mos se ious h ea s o coas al
in as uc u e.
51
Using he me hodology desc ibed in he p eceding h ee sec ions, he ollowing Code 8
was de eloped:
Code 8: De elopmen o he sec ions accoun ing o he ailu e p obabili ies, eliabili y index and
con e gence o he ob ained p obabili ies.
%% 9. Failu e, Realiabili y Index and Con e gence wi h espec o Num. o
Repe i ions
%% 9.1. Highe C i ical Heigh
%% 9.1.1. Analysis o Highe C i ical Heigh
ole ance = 0.0001; % Con e gence ole ance
window_size = 200; % Window size o s abili y
s ep_size = 50; % S ep size o sampling
max_samples = num_ epe i ions;
sample_sizes = s ep_size:s ep_size:max_samples;
% Ini ialize a ays
con e gence_ o al_high = ze os(leng h(sample_sizes), leng h(yea s),
leng h(scena ios));
con e gence_poin s_high = ze os(leng h(yea s), leng h(scena ios));
ailu e_p ob_ o al_high = ze os(leng h(yea s), leng h(scena ios));
eliabili y_indices_high = ze os(leng h(yea s), leng h(scena ios));
% Calcula e base ailu e p obabili y and eliabili y index o high
c i ical heigh
base_ ailu e_high = sum( o al_wa e _le el >= c i ical_heigh _high) /
num_ epe i ions;
ma gin_sa e y_high = c i ical_heigh _high - o al_wa e _le el;
be a_base_high = mean(ma gin_sa e y_high, 'omi nan') /
s d(ma gin_sa e y_high, 'omi nan');
% Compu ing Con e gence, ailu e p obabili ies, and eliabili y indices
o i = 1:leng h(yea s)
o j = 1:leng h(scena ios)
adjus ed_wa e _le el = o al_wa e _le el + sl _scena ios(i,j);
% Failu e p obabili y
ailu es = sum(adjus ed_wa e _le el >= c i ical_heigh _high);
ailu e_p ob_ o al_high(i,j) = ailu es / num_ epe i ions;
% Reliabili y index
ma gin_sa e y = c i ical_heigh _high - adjus ed_wa e _le el;
mean_ma gin = mean(ma gin_sa e y, 'omi nan');
s d_ma gin = s d(ma gin_sa e y, 'omi nan');
eliabili y_indices_high(i,j) = mean_ma gin / s d_ma gin;
52
3.1.11. Applica ion o he model: The Eb o Del a Sys em
Figu e 13: Top View o he Eb o Del a. Sou ce: El Nacional. (2023, Da e). La NASA dedica un es udio al
Del a de l'Eb e y ale a de su e oceso. El Nacional. h ps://www.elnacional.ca /es/sociedad/nasa-dedica-
es udio-del a-eb e-ale a- e ocede_688724_102.h ml
% Con e gence
o k = 1:leng h(sample_sizes)
n = sample_sizes(k);
ailu es_n = sum(adjus ed_wa e _le el(1:n) >=
c i ical_heigh _high);
con e gence_ o al_high(k,i,j) = ailu es_n / n;
end
% Find con e gence poin
o k = window_size:leng h(sample_sizes)
window = con e gence_ o al_high((k-window_size+1):k,i,j);
i s d(window) < ole ance
con e gence_poin s_high(i,j) = sample_sizes(k-
window_size+1);
b eak;
end
end
end
end
53
The analy ical amewo k p esen ed he e add esses he complex coas al en i onmen
o he Eb o Del a, ea u ing a 50-kilome e coas line cha ac e ized by low-lying
opog aphy (Figu e 13), whe e a ious na u al p ocesses in e ac o shape i s ex ensi e
beach sys ems.
The amewo k conside s key en i onmen al p ocesses ha egula e coas al beha iou
h oughou he del a. These include Medi e anean wa e pa e ns and seasonal s o m
su ge e en s. Sedimen dynamics, his o ically d i en by he Eb o Ri e bu now modi ied
by ups eam dam cons uc ion, play a c ucial ole in beach e olu ion. The sys em aces
cu en en i onmen al challenges, including e osion in exposed a eas, sea-le el ise
ulne abili y, and na u al subsidence.
This code implemen a ion de elops upon hese undamen al cha ac e is ics,
es ablishing a s anda dized app oach o e alua ing indi idual beaches wi hin he del a.
While main aining consis ency in he me hodology, i allows o he conside a ion o
local a ia ions in mo phodynamical condi ions. The e alua ion o beaches o e he Eb o
Del a sys em equi es he analysis o h ee undamen al pa ame e s ha con ol hei
beha iou and ulne abili y o coas al haza ds. These pa ame e s o m he ounda ion
o he analy ical amewo k and a e essen ial o unde s anding he dynamic esponse
o hese coas al sys ems.
The beach slope se es as a p ima y indica o o coas al ulne abili y and wa e-beach
in e ac ion. This geome ic cha ac e is ic de e mines how wa e ene gy is dissipa ed
ac oss he beach p o ile and in luences he po en ial o looding e en s. Fo beaches in
he Ta agona egion and speci ically he Eb o Del a, his da a is ob ained h ough he
Coas al Viewe pla o m (Llib e Ve d) om he Ca og aphic and Geological Ins i u e o
Ca alonia [41]. By selec ing he speci ic beach o in e es , de ailed slope in o ma ion
can be ex ac ed, p o iding a comple e unde s anding o he beach p o ile
con igu a ion.
Wa e incidence di ec ions cons i u e he second c i ical pa ame e in he analysis
amewo k. These di ec ional componen s desc ibe he p edominan angles a which
wa es app oach he coas line. The analysis o wa e di ec ions is conduc ed using
ca og aphic ools, whe e a e e ence line is es ablished o he beach o in e es . F om
he eigh possible ca dinal and in e ca dinal di ec ions, i e p ima y wa e app oach
di ec ions a e iden i ied as po en ially a ec ing he beach. This di ec ional dis ibu ion
signi ican ly in luences sedimen anspo pa e ns along he sho e and de e mines he
p obabili y o wa e impac di ec ions on di e en beaches.
The hi d essen ial pa ame e is he c i ical wa e heigh , which ep esen s a c ucial
h eshold pa ame e in beach looding analysis, pa icula ly signi ican in low-lying a eas
such as he Eb o Del a. This pa ame e , ob ained h ough he VISSIR [40] isualiza ion
pla o m o he Ca og aphic and Geological Ins i u e o Ca alonia, helps iden i y he
highes poin s ha se e as he las line o de ence agains looding. When loodwa e
b eaches his highes poin , i can apidly low o lowe ele a ions and inland a eas,
po en ially causing beach e osion and in as uc u e damage.

54
Howe e , a comp ehensi e looding assessmen equi es conside a ion o bo h he
highes ele a ions and lowe -lying a eas. While he highes poin ac s as a c i ical
h eshold o looding p o ec ion, lowe -lying sec ions can unc ion as channels di ec ing
wa e owa d inne a eas, e en be o e he highes poin is o e opped. This combined
analysis o bo h high and low poin s, all accessible h ough VISSIR, enables a de ailed
unde s anding o looding mechanisms and ulne abili ies.
Toge he , hese h ee pa ame e s es ablish a comp ehensi e amewo k o analysing
beach beha iou wi hin he Eb o Del a sys em. The Ma quesa and The T abucado
beaches we e speci ically selec ed o his analysis due o hei con as ing posi ions
wi hin he del a, each loca ed on opposi e sides o he sys em, demons a ing how local
cha ac e is ics in luence beach beha iou and ulne abili y o coas al p ocesses.
The Ma quesa, loca ed in he no he n hemidel a, and The T abucado , posi ioned in
he sou he n sec ion, expe ience dis inc wa e pa e ns and mo phodynamical
condi ions.
3.1.12. Beach Cha ac e is ics: Si e-Speci ic Analysis o The Ma quesa Beach
Loca ed in he no he n hemidel a o he Eb o Del a sys em (Figu e 14), Ma quesa Beach
(Figu e 15) ep esen s a coas al en i onmen wi h a cha ac e is ic p o ile o ine golden
sand composi ion, c ea ing a mode a e slope ha in luences wa e un-up p ocesses. I s
posi ion makes i pa icula ly suscep ible o no he ly wind pa e ns, speci ically he
T amon ana, which plays a signi ican ole in sedimen anspo dynamics.
The beach sys em inco po a es a well-de eloped be m and expe iences ela i ely
shallow wa e s, con ibu ing o i s speci ic wa e ans o ma ion pa e ns. The back-
beach a ea includes bo h na u al and modi ied zones, measu ing c i ical wa e heigh s is
pa icula ly ele an o lood isk managemen . I s o ien a ion allows o wa e
app oach om mul iple di ec ions, hough i emains pa ially shel e ed om ce ain
wa e angles by i s posi ion wi hin he del a con igu a ion, howe e in his s udy, as he
inpu da a was ob ained om buoys loca ed in deep wa e all possible di ec ions we e
conside ed, o conside he all he ac ual p obable cases.
55
Figu e 14: Expanded op iew o The Ma quesa. Sou ce: Google maps.
Figu e 15: Top View and O ien a ion o he Ma quesa Beach: Sou ce: Google Maps.
56
- The Ma quesa Beach Slope Analysis
The Ma quesa beach slope alue is ob ained h ough he Coas al Viewe pla o m (Llib e
Ve d) om he Ca og aphic and Geological Ins i u e o Ca alonia [41]. The pla o m
p o ides de ailed opog aphic and ba hyme ic da a ha allows o p ecise slope
calcula ions. Analysis o he beach p o ile shows ha The Ma quesa has a slope o 0.05
(5%). This measu emen e lec s he beach's cha ac e is ic gen le g adien , which is
ypical o i s posi ion in he no he n hemidel a. The slope calcula ion conside s he
p o ile om he sho eline ex ending o sho e, p o iding an a e age g adien ha is
ep esen a i e o he beach ace used in wa e un-up calcula ions.
- The Ma quesa Beach Wa e Exposu e
Gi en The Ma quesa beach's loca ion in he Eb o Del a, as i is de e mined in Figu e 15,
wa es can app oach om he ollowing p incipal di ec ions: NW (No hwes ), N (No h),
NE (No heas ), E (Eas ), and SE (Sou heas ). The mos signi ican wa e ac ion ypically
comes om he eas e ly (E) and no heas e ly (NE) di ec ions, as hese wa es a i e
di ec ly om he Medi e anean Sea. The beach also expe iences wa e pa e ns om
he sou heas e n (SE) di ec ion, pa icula ly du ing ce ain me eo ological condi ions.
- C i ical Heigh Analysis o The Ma quesa Beach
Fo The Ma quesa Beach, he c i ical heigh has been es ablished a 2.2 me es,
ep esen ing he highes ele a ion poin ha se es as he las de ence agains looding.
This h eshold was de e mined based on he beach's opog aphic cha ac e is ics,
including i s mode a e slope and ela i ely wide beach p o ile, which p o ides some
na u al p o ec ion agains wa e ac ion. Howe e , a comp ehensi e looding analysis
e eals ha he inne a eas become ulne able a a much lowe h eshold o 1.3 me es,
whe e lowe -lying sec ions can unc ion as channels di ec ing wa e owa d ag icul u al
a eas and in as uc u e.
Th ough opog aphic da a ob ained om VISSIR (Figu e 16), i is possible o obse e how
hese di e en ele a ion poin s in e ac du ing looding e en s. When wa es exceed he
lowe 1.3-me e h eshold, wa e can begin in il a ing h ough hese na u al channels,
po en ially a ec ing inland a eas e en be o e he highes poin is o e opped. Once
wa es su pass he 2.2-me e c i ical heigh , he en i e beach sys em becomes
ulne able o mo e se e e looding, wi h apid inunda ion o lowe a eas and po en ial
damage o nea by in as uc u e.
57
Figu e 16: C i ical heigh and low ele a ions o he Ma quesa Beach . Sou ce: Ins i u Ca og à ic i
Geològic de Ca alunya. VISSIR3 - Visualizado de in o mación geog á ica. h p://s .icgc.ca / issi 3/
This dual h eshold conside a ion demons a es he impo ance o analysing bo h high
and low poin s in lood isk assessmen . While he 2.2-me e c i ical heigh p o ides a
conse a i e measu e o maximum p o ec ion, he lowe 1.3-me e h eshold ale s us
o ini ial looding isks h ough na u al channels. This comp ehensi e unde s anding
enables mo e e ec i e lood isk managemen s a egies, which a e pa icula ly c ucial
o p o ec ing he adjacen ag icul u al a eas and in as uc u e ha cha ac e ize
Ma quesa Beach's back-beach en i onmen .
In his way, he local cha ac e is ics o he Ma quesa a e de ined in Code 9.
Code 9: De ini ion o he mo phological cha ac e is ics and wa e exposu e di ec ions o he Ma quesa.
%% The Ma quesa
slope = 0.05; % Beach slope
c i ical_heigh _high = 2.2; % Highe c i ical wa e le el (m)
c i ical_heigh _low = 1.3; % Lowe c i ical wa e le el (m)
alid_di ec ions = {'NW', 'N', 'NE', 'E', 'SE'};
64
4. Resul s
The esul s sec ion p esen s he analy ical ou comes ob ained h ough he p e iously
desc ibed modelling amewo k. The analysis ocuses on wo dis inc coas al loca ions:
The Ma quesa and The T abucado beaches, whe e mul iple isualiza ion se s ha e been
gene a ed o illus a e a ious aspec s o coas al ulne abili y unde di e en scena ios.
The assessmen in ol es a comple e analysis o si e-speci ic esponses gene a ed by he
Mon e Ca lo simula ion model. The model was execu ed o 1 million i e a ions using
MATLAB so wa e on an HP Pa ilion compu e , wi h each si e equi ing nea ly one hou
o compu a ion. While main aining consis en compu a ional pa ame e s and modi ying
he cha ac e is ics o each loca ion. The analysis amewo k enables di ec a ibu ion
o obse ed a ia ions o he dis inc i e mo phological ea u es and di ec ional
exposu e pa e ns unique o each beach loca ion.
The analy ical esul s demons a e wa e le el p ojec ions unde a ious clima e change
scena ios, inco po a ing bo h nea - e m and long- e m o ecas s, while accoun ing o
local subsidence e ec s. These p ojec ions a e complemen ed by de ailed p obabili y o
ailu e assessmen s, which ha e been e alua ed h ough wo dis inc app oaches:
conside ing he comple e da ase o samples and speci ically ocusing on s o m e en
condi ions. Addi ionally, he analysis includes con e gence s udies and a comp ehensi e
eliabili y assessmen ha e alua es sa e y le els agains looding du ing s o m e en s,
p o iding quan i a i e measu es o each beach's esilience unde ex eme condi ions.

65
4.1. The Ma quesa
4.1.1. Wa e analysis
Figu e 21: Hs Dis ibu ion o he Ma quesa in unc ion o he clima e. Sou ce: Main.
The igu e shown abo e (Figu e 21) p esen s a compa a i e analysis o wa e heigh
dis ibu ions a The Ma quesa beach, illus a ing he impac o di ec ional and s o m
e en s il e ing c i e ia on signi ican wa e heigh s (Hs).
The Ini ial Hs Dis ibu ion con ains he comple e wa e heigh da ase , anging om 0 o
10 me es, wi h mos o he wa e heigh s concen a ed be ween 0 and 2 me es. This
dis ibu ion ep esen s all eco ded wa e condi ions, achie ing 100% alid samples, and
demons a es mo e alues on he le and a long ail on he igh , which is a pa e n
ypical o wa e heigh dis ibu ions in he Medi e anean egion.
The Final Hs Dis ibu ion ocuses exclusi ely on s o m e en s, ep esen ing only 0.5% o
he o al eco ded cases. These e en s a e speci ically il e ed o include wa es
app oaching om NW, N, NE, E, and SE di ec ions wi h heigh s equal o o exceeding 3
me es om he Ini ial Hs Dis ibu ion. This il e ed da ase has been ecalcula ed o
analyse ex eme condi ions wi hin hese speci ic di ec ional cons ain s, wi h wa e
heigh s p ima ily anging be ween 3 and 6 me es. The esul ing dis ibu ion is
a e wa ds u ilized o un-up compu a ions, ensu ing ha he analysis cap u es he
mos c i ical condi ions ha in luence coas al looding isks a The Ma quesa.
66
4.1.2. Wa e Le el Componen s
Figu e 22: Wa e Le el Componen s o The Ma quesa beach. Sou ce: Main.
The igu e showed abo e (Figu e 22), illus a es he key componen s con ibu ing o
wa e le el a ia ions in cu en scena io a The Ma quesa beach h ough ou dis inc
dis ibu ions.
The As onomical Tide Dis ibu ion shows a pa e n anging om 0.17 o 0.53 me es,
wi h peak alues concen a ed a ound 0.34 me es, ela ed wi h he mean sea le el o
33.37cm. The Me eo ological Residue Dis ibu ion desc ibes a Gaussian unc ion
spanning om 0.30 o 0.70 me es, cen ed a 0.50 me es, e lec ing he a mosphe ic
p essu e and wind in luences on wa e le els.
Wa e Run-up (R2%), conside ing Hs il e ed o s o m e en s alues in unc ion o hei
di ec ion, exhibi s again a igh -skewed dis ibu ion wi h mos e en s concen a ed
be ween 0.5 and 0.8 me es, declining owa d highe alues.
The To al Wa e Le el Dis ibu ion, in eg a ing all componen s, anges om 1.2 o 2.6
me es. No ably, a signi ican po ion o he dis ibu ion exceeds he Low C i ical Heigh
(1.30m), while alues abo e he High C i ical Heigh (2.20m) a e in equen , sugges ing
mode a e ulne abili y o ex eme wa e le el e en s a The Ma quesa beach.
67
4.1.3. Failu e P obabili y du ing S o m E en s
Figu e 23: Failu e P obabili ies In Case o S o m E en by C i ical Heigh ( he Ma quesa). Sou ce: Main.
Figu e 23, illus a es he ailu e p obabili ies (0 o 1) o The Ma quesa beach du ing
s o m e en s ac oss di e en clima e change scena ios and ime ho izons, analysing
bo h he high (2.2m) and low (1.3m) c i ical heigh s.
Fo he high c i ical heigh , ep esen ing he beach's ele a ed a eas, he cu en scena io
shows minimal ailu e p obabili y (0.002) in 2025. O cou se, p obabili y alues appea
iden ical o all h ee scena ios (Cu en , RCP4.5, and RCP8.5) in 2025, as he e ec s o
di e en clima e change ajec o ies ha e no ye had ime o ma e ialize in his nea -
e m ime ame. This isk emains ela i ely low in 2050, wi h sligh inc eases o 0.022
and 0.030 o RCP4.5 and RCP8.5 scena ios, espec i ely. Howe e , by 2100, he
p obabili y o ailu e inc eases subs an ially, eaching 0.440 unde RCP4.5 and 0.999
unde RCP8.5, indica ing a signi ican long- e m ulne abili y when accoun ing o
subsidence e ec s and clima e change scena ios.
Rega ding he low c i ical heigh , co esponding o he beach's lowe sec ions, he
analysis e eals consis en ly high ailu e p obabili ies ac oss all scena ios and
ime ames. S a ing wi h 0.932 in 2025 o all scena ios, he p obabili y inc eases o
0.991-1.000 by 2050 and main ains hese ele a ed le els h ough 2100. These esul s
demons a e ha du ing s o m e en s, he lowe sec ions o The Ma quesa Beach ace
immedia e and pe sis en looding isks, wi h clima e change and subsidence e ec s
ampli ying his ulne abili y o e ime.
68
The explici con as be ween high and low c i ical heigh s emphasizes he spa ial
a iabili y o looding isks wi hin Ma quesa Beach, while he empo al p og ession
ac oss scena ios highligh s he inc easing impac o clima e change and subsidence on
coas al looding p obabili ies.
4.1.4. Wa e Le el Exceedance
The wa e le el exceedance p obabili y g aph o Ma quesa Beach illus a es an
inc easing lood isk pa e n o e ime (Figu e 24). Cu en condi ions in 2025 show
ela i ely low p obabili ies o exceeding he high c i ical heigh (2.2m). Howe e , bo h
mode a e (RCP4.5) and se e e (RCP8.5) clima e change scena ios indica e p og essi ely
highe wa e le els by 2050, wi h subs an ial inc eases p ojec ed o 2100. The low
c i ical heigh (1.3m) emains consis en ly ulne able ac oss all ime ames and
scena ios. Mos no ably, he RCP8.5 scena io o 2100 demons a es a signi ican
p obabili y o exceedingly e en he high c i ical h eshold, indica ing se e e u u e
looding isks wi hou in e en ion.
Figu e 24: Wa e Le el Exceedance In Case o S o m E en by C i ical Heigh ( he Ma quesa). Sou ce:
Main.
4.1.5. Con e gence Analysis
This sec ion de elops he con e gence analysis o he ailu e p obabili y wi h espec o
all he samples s udied. The Ma quesa beach's high c i ical heigh (2.20m) e eals
dis inc pa e ns ac oss di e en clima e scena ios, wi h con e gence de e mined
h ough speci ic c i e ia: a ole ance o 0.0001, a s abili y window o 200 samples, and
p og essi e e alua ion a 50-sample in e als (Figu e 24).
69
Figu e 25: Con e gence Analysis-Cu en Scena io, High C i ical Heigh ( he Ma quesa). Sou ce: Main.
Unde he cu en scena io, as shown on he igu e abo e (Figu e 25), he analysis
demons a es e icien con e gence wi h minimal ailu e p obabili y. The model
achie es s abili y a n=50 samples, mee ing he con e gence c i e ia wi h consis en
e alua ions ac oss he speci ied window size, while main aining a no ably low ailu e
p obabili y (P =0.0000) h ough 2025 and 2050. Fo 2100, a sligh inc ease in ailu e
p obabili y is obse ed (P =0.0001), hough s ill con e ging e icien ly wi hin he same
sampling pa ame e s.
Figu e 26: Con e gence Analysis-RCP4.5 Scena io, High C i ical Heigh ( he Ma quesa). Sou ce: Main.

70
The RCP4.5 scena io, shown on Figu e 26, exhibi s mo e complex con e gence
beha iou . While 2025 and 2050 main ain quick con e gence a n=50 samples wi h low
ailu e p obabili ies (P =0.0000 and 0.0001 espec i ely), he 2100 p ojec ion equi es
subs an ially mo e samples (n=11700) o sa is y he con e gence c i e ia. This inc eased
sample equi emen , yielding a ailu e p obabili y o 0.0022, e lec s he g owing
unce ain y in long- e m p ojec ions, necessi a ing mo e i e a ions o achie e s abili y
wi hin he speci ied ole ance.
Figu e 27: Con e gence Analysis-RCP8.5 Scena io, High C i ical Heigh ( he Ma quesa). Sou ce: Main.
The RCP8.5 scena io (Figu e 27) p esen s he mos challenging con e gence pa e ns.
While ea ly pe iods main ain e icien con e gence like o he scena ios, he 2100
p ojec ion demands he highes numbe o samples (n=11900) o each s abili y wi hin
he de ined ole ance and window size, esul ing in a ailu e p obabili y o 0.0049. The
con e gence pa e n shows ini ial luc ua ions be o e s abilizing ac oss he speci ied
window, indica ing inc eased complexi y in modelling mo e se e e clima e change
impac s unde he s ic con e gence c i e ia es ablished.
This p og essi e inc ease in equi ed samples and ailu e p obabili ies ac oss scena ios
and ime ames demons a es he g owing complexi y and unce ain y in modelling
mo e se e e clima e change impac s, pa icula ly o long- e m p ojec ions, while
ensu ing obus esul s h ough igo ous con e gence c i e ia.
The con e gence analysis o Ma quesa Beach's low c i ical heigh (1.30m)
demons a es a no ably di e en pa e n compa ed o he high c i ical heigh , wi h
consis en beha iou ac oss all clima e scena ios while adhe ing o he es ablished
con e gence c i e ia ( ole ance o 0.0001, s abili y window o 200 samples, and 50-
sample in e als).
71
Figu e 28: Con e gence Analysis-Cu en Scena io, Low C i ical Heigh ( he Ma quesa). Sou ce: Main.
Figu e 29: Con e gence Analysis-RCP4.5 Scena io, Low C i ical Heigh ( he Ma quesa). Sou ce: Main.
72
Figu e 30: Con e gence Analysis-RCP8.5 Scena io, Low C i ical Heigh ( he Ma quesa). Sou ce: Main.
Fo all scena ios (Cu en , RCP4.5, and RCP8.5), he analysis equi es a signi ican ly
highe numbe o samples (n=11900) o achie e con e gence, indica ing g ea e
complexi y in modelling he lowe ele a ion beha iou . The ailu e p obabili ies show a
p og essi e pa e n, s a ing a P =0.0046 o 2025 and inc easing sligh ly o P =0.0049
o bo h 2050 and 2100 (Figu e 28, 29 and 30, espec i ely).
The con e gence pa e ns exhibi ini ial luc ua ions be o e s abilizing, pa icula ly
e iden in he i s 100000 samples. A e his poin , all scena ios demons a e
ema kable s abili y, main aining consis en ailu e p obabili ies ac oss he emainde
o he simula ion pe iod.
This s abili y is pa icula ly no able as i pe sis s ac oss di e en clima e scena ios,
sugges ing ha he low-lying a eas' ulne abili y is mo e in luenced by hei opog aphic
posi ion han by he speci ic clima e change scena io.
The succeeding con e gence analysis examines looding unde c i ical s o m condi ions,
ocusing on wa es ha exceed heigh h esholds and app oach om key di ec ions. This
assessmen o ex eme e en s demons a es he Mon e Ca lo simula ion's obus ness
in e alua ing beach pe o mance du ing se e e condi ions. The model achie es s able
p obabili y calcula ions e en when analysing his speci ic subse o haza dous scena ios,
alida ing i s e ec i eness o coas al ulne abili y assessmen . While he highe
ele a ions showed inc easing complexi y and highe ailu e p obabili ies wi h mo e
se e e clima e scena ios, he lowe ele a ions main ain consis en beha iou , indica ing
hei pe sis en ulne abili y ega dless o he clima e change ajec o y, sugges ing ha
low-lying a eas equi e immedia e a en ion ega dless o he clima e change scena io
conside ed (Figu es 31 and 32).
73
Figu e 31: Con e gence Analysis-Valid Samples High C i ical Heigh ( he Ma quesa). Sou ce: Main
Figu e 32: Con e gence Analysis-Valid Samples Low C i ical Heigh ( he Ma quesa). Sou ce: Main.
80
4.2.4. Wa e Le el Exceedance
This wa e le el exceedance p obabili y g aph o T abucado Beach e eals conce ning
ulne abili y pa e ns. Unde cu en condi ions in 2025, he beach al eady shows a
no able p obabili y o exceeding i s high c i ical heigh (2.1m). The p og ession h ough
ime demons a es a ma ked inc ease in looding isk, wi h he 2050 p ojec ions
showing signi ican igh wa d shi s in he p obabili y cu es o bo h mode a e (RCP4.5)
and se e e (RCP8.5) clima e change scena ios (Figu e 38).
The si ua ion becomes pa icula ly se e e by 2100, whe e he p obabili y cu es indica e
equen exceedance o c i ical h esholds. Mos no ably, he RCP8.5 scena io o 2100
shows ex emely high p obabili ies o wa e le els exceeding bo h c i ical heigh s,
sugges ing nea -ce ain looding condi ions du ing se e e wea he e en s. The low
c i ical heigh (1.2m) appea s consis en ly ulne able ac oss all scena ios, wi h 100%
exceedance p obabili y indica ed by he la uppe po ions o all cu es.
Compa ing his o he Ma quesa beach, he T abucado demons a es g ea e o e all
ulne abili y, wi h highe exceedance p obabili ies ac oss all scena ios and pe iods. This
inc eased ulne abili y shows clea ly how all he p obabili y cu es ha e shi ed
igh wa d, indica ing ha highe wa e le els occu mo e equen ly a his loca ion. The
analysis sugges s ha he T abucado equi es mo e immedia e a en ion o coas al
p o ec ion measu es, pa icula ly gi en i s apid de e io a ion in p o ec i e capaci y
unde u u e clima e scena ios.
Figu e 38: Wa e Le el Exceedance In Case o S o m E en by C i ical Heigh ( he T abucado ). Sou ce:
Main.

81
4.2.5. Con e gence Analysis
The con e gence analysis o T abucado Beach's high c i ical heigh (2.10m) e eals
dis inc pa e ns ac oss clima e scena ios and empo al ho izons while ollowing he
es ablished con e gence c i e ia ( ole ance o 0.0001, s abili y window o 200 samples,
and 50-sample s eps).
Figu e 39: Con e gence Analysis-Cu en Scena io, High C i ical Heigh ( he T abucado ). Sou ce: Main.
Unde he cu en scena io (Figu e 39), he analysis demons a es a p og essi e inc ease
in equi ed samples o con e gence. Fo 2025, con e gence is achie ed e icien ly a
n=50 samples wi h a ailu e p obabili y o 0.0002. The 2050 p ojec ion equi es n=1800
samples o each s abili y wi h P =0.0003, while 2100 demands n=8650 samples o
achie e con e gence wi h P =0.0007, indica ing inc easing complexi y in long- e m
p ojec ions.
82
Figu e 40: Con e gence Analysis-RCP4.5 Scena io, High C i ical Heigh ( he T abucado ). Sou ce: Main.
The RCP4.5 scena io (Figu e 40) shows mo e p onounced changes in con e gence
beha iou . While main aining e icien con e gence o 2025 (n=50, P =0.0002), he
2050 p ojec ion equi es n=5100 samples o achie e s abili y wi h P =0.0013. The 2100
ho izon exhibi s he highes complexi y, needing n=16650 samples o con e ge wi h
P =0.0066, e lec ing inc eased unce ain y in long- e m clima e change impac s.
Figu e 41: Con e gence Analysis-RCP8.5 Scena io, High C i ical Heigh ( he T abucado ). Sou ce: Main.
83
The RCP8.5 scena io shown in Figu e 41, main ains he same con e gence
cha ac e is ics o 2025 and 2100 as RCP4.5, bu shows dis inc beha iou o 2050,
equi ing n=9050 samples o con e gence wi h P =0.0018. This inc eased sample
equi emen o mid-cen u y p ojec ions sugges s complexi y in modelling he ansi ion
pe iod unde mo e se e e clima e change condi ions.
The con e gence analysis o T abucado Beach's low c i ical heigh (1.20m)
demons a es consis en beha iou ac oss all clima e scena ios and empo al ho izons,
ollowing he es ablished con e gence c i e ia ( ole ance o 0.0001, s abili y window o
200 samples, and 50-sample s eps).
Figu e 42: Con e gence Analysis-Cu en Scena io, Low C i ical Heigh ( he T abucado ). Sou ce: Main.
Figu e 43: Con e gence Analysis-RCP4.5 Scena io, Low C i ical Heigh ( he T abucado ). Sou ce: Main.
84
Figu e 44: Con e gence Analysis-RCP8.5 Scena io, Low C i ical Heigh ( he T abucado ). Sou ce: Main.
No ably, all scena ios (Cu en , RCP4.5, and RCP8.5) and ime ames (2025, 2050, and
2100) exhibi iden ical con e gence cha ac e is ics, equi ing n=16650 samples o
achie e s abili y wi h a ailu e p obabili y o 0.0066. This consis ency sugges s ha he
low-lying a eas o The T abucado main ain simila ulne abili y pa e ns ega dless o
he clima e change scena io o empo al ho izon conside ed (Figu es 42, 43 and 44,
espec i ely).
The con e gence pa e ns show ini ial luc ua ions in he i s 100000 samples be o e
s abilizing, wi h all scena ios main aining s eady ailu e p obabili ies he simula ion
pe iod. This s abili y is ema kable as i pe sis s ac oss di e en clima e scena ios and
ime ames.
By analysing exclusi ely s o m e en condi ions, o de e mine he p obabili y o ailu e,
whe e wa e heigh s exceed es ablished h esholds and app oach om alid di ec ions,
he con e gence analysis s ill demons a es obus s a is ical s abili y. The analysis
con i ms ha e en when cons aining he da ase o hese ex eme e en s, he Mon e
Ca lo simula ion achie es eliable con e gence, alida ing he me hodology's
e ec i eness in e alua ing beach pe o mance du ing s o ms. So, he con e gence
assessmen o his subse o da a shows ha he p obabili y o ailu e calcula ions
main ains s abili y despi e he educed sample size.
85
Figu e 45: Con e gence Analysis-Valid Samples High C i ical Heigh ( he T abucado ). Sou ce: Main
Figu e 46: Con e gence Analysis-Valid Samples Low C i ical Heigh ( he T abucado ). Sou ce: Main

86
Simila o wha happened in The Ma quesa, highe ele a ions show inc easing ailu e
p obabili ies unde mo e se e e clima e scena ios, which s abilize a e he i s 100000
samples, as shown in Figu e 45, while The T abucado 's low-lying a eas demons a e he
wo s uni o m beha iou (Figu e 46). This sugges s ha The T abucado 's lowe
ele a ions p esen mo e consis en ulne abili y cha ac e is ics, main aining s eady
ailu e p obabili ies ac oss all scena ios and ime ames. This con as s wi h The
Ma quesa's mo e a iable con e gence pa e ns and gene ally lowe ailu e
p obabili ies, pa icula ly in ea lie ime ames and less se e e clima e scena ios.
4.2.6 Reliabili y Analysis
Figu e 47: Reliabili y Indices by Scena io. High C i ical Heigh o he T abucado . Sou ce: Main.
The eliabili y indices o T abucado Beach e eal awa eness o i s ulne abili y o
looding, wi h dis inc pa e ns eme ging ac oss di e en ele a ions and ime ho izons
(Figu e 47).
Fo he high c i ical heigh (2.10m), he ini ial mode a e eliabili y index o 2.45 in 2025
indica es ha e en he ele a ed a eas o The T abucado s a wi h only accep able
sa e y le els agains looding du ing s o m e en s. This con as s sha ply wi h The
Ma quesa's obus ini ial index o 4.83, sugges ing ha The T abucado 's highe
ele a ions a e inhe en ly mo e ulne able.
87
The apid de e io a ion by 2050, pa icula ly unde clima e change scena ios (d opping
o 0.59 and 0.31), indica es a ansi ion owa d unsa e condi ions much ea lie han The
Ma quesa. By 2100, he nega i e indices (-1.70 and -3.41 unde RCP scena ios) signi y
ha e en hese highe ele a ions will likely expe ience egula looding du ing s o ms,
ep esen ing a undamen al shi in he beach's p o ec i e capaci y.
Figu e 48: Reliabili y Indices by Scena io. Low C i ical Heigh o he T abucado . Sou ce: Main.
The low c i ical heigh (1.20m) analysis e eals e en mo e conce ning implica ions. The
s ongly nega i e ini ial indices a ound -2.70 in 2025 indica e ha hese a eas a e
al eady in a c i ical s a e, wi h looding being a nea -ce ain y du ing s o m e en s. The
p og ession o ex eme nega i e alues by 2100 (-6.85 and -8.56 unde RCP scena ios)
sugges s hese lowe a eas will become essen ially uninhabi able du ing s o m
condi ions, equi ing immedia e and signi ican adap a ion s a egies (Figu e 48).
The con as be ween The T abucado and The Ma quesa's eliabili y indices emphasizes
how local coas al cha ac e is ics signi ican ly in luence ulne abili y. The T abucado 's
mo e se e e indices ac oss all scena ios can be a ibu ed o i s speci ic mo phological
ea u es, including i s wide di ec ional exposu e o wa es and s eepe beach slope.
These cha ac e is ics make i no only mo e suscep ible o immedia e looding isks bu
also mo e sensi i e o he combining e ec s o sea-le el ise and clima e change,
sugges ing a need o mo e u gen and comp ehensi e adap a ion s a egies compa ed
o The Ma quesa.
88
5. Conclusions
Based on he comp ehensi e analysis o The Ma quesa and The T abucado beaches in
he Eb o Del a, se e al key insigh s eme ge conce ning hei e ec i eness as na u al
coas al de ences. The indings e eal dis inc pa e ns in how beach mo phology and
o ien a ion in luence p o ec i e capaci y, wi h signi ican implica ions o coas al
managemen s a egies.
The esea ch es ablishes ha geome ic cha ac e is ics undamen ally de e mine a
beach's de ensi e capabili ies. While bo h s udy si es expe ience iden ical idal and
me eo ological condi ions, hei esponses o wa e ac ion di e signi ican ly due o hei
dis inc con igu a ions. The Ma quesa's highe ele a ions cu en ly demons a e obus
p o ec ion agains looding, hough his e ec i eness is p ojec ed o diminish g adually
h ough mid-cen u y be o e acing subs an ial challenges by 2100 unde se e e clima e
scena ios. In con as , The T abucado exhibi s inc eased sensi i i y o en i onmen al
o ces e en a i s ele a ed sec ions, wi h p o ec ion le els s a ing a accep able bu
conce ning alues and con i ming accele a ed de e io a ion unde clima e change
p ojec ions.
The analysis iden i ies beach ele a ion as he c i ical de e minan o lood p o ec ion
capabili y. The lowe sec ions o bo h beaches al eady demons a e signi ican
ulne abili y and show pe sis en suscep ibili y o looding ac oss all clima e scena ios,
indica ing ha ele a ion challenges cons i u e a undamen al conce n independen o
clima e change ajec o ies.
E en hough, The T abucado 's si ua ion is pa icula ly e en mo e se ious han he
Ma quesa’s, which equi es mo e immedia e a en ion o coas al p o ec ion measu es,
pa icula ly gi en i s apid de e io a ion in p o ec i e capaci y unde u u e clima e
scena ios.
The me hodology's eliabili y is alida ed h ough ex ensi e s a is ical es ing,
demons a ing consis en esul s ac oss a ying sample sizes while accoun ing o he
inc easing complexi y o long- e m clima e p ojec ions. This obus analy ical amewo k
p o ides ai h ul e alua ions o bo h cu en and u u e scena ios, hough i s alidi y
depends on main aining assumed beach geome ies h ough egula main enance
p og ams.
When ex ended o he gene al Eb o Del a sys em, he indings sugges ha beaches wi h
simila physical cha ac e is ics will encoun e compa able challenges. No he n- acing
beaches bene i om na u al p o ec ion agains p edominan s o m di ec ions, while
hose wi h b oade di ec ional exposu e ace mo e immedia e isks equi ing equen
in e en ion. The consis en ulne abili y pa e ns obse ed a lowe ele a ions
h oughou he s udy a ea emphasize he need o immedia e a en ion o low-lying
sec ions ac oss he del a.
89
These conclusions unde sco e he u gen equi emen o a comp ehensi e coas al
managemen s a egy ha in eg a es bo h main enance and adap a ion componen s.
A eas below 1.30 me es demons a e immedia e ulne abili y ega dless o loca ion,
while he accele a ing de e io a ion o p o ec i e capaci y unde clima e change
scena ios demands adap i e managemen conside a ions. This s a egy mus add ess
bo h immedia e ulne abili ies and long- e m clima e impac s h ough con inuously
adjus ed main enance p og ams.
Looking ahead, hese indings es ablish a scien i ic ounda ion o e idence-based
coas al p o ec ion s a egies. The analysis demons a es ha main aining cu en beach
geome ies alone will no be enough, s a egic in e en ions o enhance p o ec i e
ea u es, pa icula ly h ough ele a ion modi ica ions, will be essen ial o long- e m
coas al esilience. As clima e change con inues o eshape coas al dynamics, he
implemen a ion o hese insigh s becomes inc easingly c i ical o p ese ing he Eb o
Del a's na u al and cul u al he i age.
The message om coas al egions g ows clea e and mo e u gen each day. As wa e s
ise and s o ms in ensi y, he ea ly signs o clima e change's impac on coas lines
become inc easingly appa en . The Eb o Del a's beaches ell a s o y ha coas al
communi ies wo ldwide need o hea , hey se e as an ea ly wa ning sys em,
demons a ing he consequences when na u al de ences begin o ail.
The esea ch lea es no oom o doub , wi hou immedia e ac ion o p o ec and
s eng hen hese na u al ba ie s, he losses ex end a beyond sand and sho eline. A
conce n a e es ablished communi ies, ich ecosys ems ha ha e been he e o
cen u ies, and cul u al he i age ha , once claimed by he sea, emains i eco e able.
This momen s ands as a c i ical c isis. The ad ancing wa es will no pause o ex ended
delibe a ion o delayed decisions. The scien i ic e idence s ands clea , he da a is in
on o us, and he pa h o wa d demands immedia e ac ion. Tomo ow's ising ides
ad ance ine i ably, and he esilience o coas al egions depends en i ely on ac ions
aken in he immedia e e m.
96
[44] Uni ed Na ions. (2015). Goal 13: Take u gen ac ion o comba clima e change and
i s impac s. In T ans o ming ou wo ld: The 2030 agenda o sus ainable de elopmen .
Re ie ed om h ps://sdgs.un.o g/goals/goal13
[45] In e na ional Ene gy Agency. (2019). Global ene gy & CO2 s a us epo 2019.
h ps://www.iea.o g/ epo s/global-ene gy-co2-s a us- epo -2019

97
Appendices
1. Model code
clea all;
close all;
%% 1. Cha ac e is ics o he model
%load('base_mon e_ca lo.ma ');
load('my_ abucado .ma ')
%% The Ma quesa
% slope = 0.05; % Beach slope
% c i ical_heigh _high = 2.2; % Highe c i ical wa e le el (m)
% c i ical_heigh _low = 1.3; % Lowe c i ical wa e le el (m)
% alid_di ec ions = {'NW', 'N', 'NE', 'E', 'SE'};
%% The T abucado
slope = 0.065; % Beach slope
c i ical_heigh _high = 2.1; % Highe c i ical wa e le el (m)
c i ical_heigh _low = 1.2; % Lowe c i ical wa e le el (m)
alid_di ec ions = {'NE', 'E', 'SE','S','SW'};
%% Gene al ca ac e is ics
o al_hou s = 8760; % To al hou s in a non-leap yea
num_ epe i ions = 1000000; % Numbe o epe i ions
g = 9.81; % G a i a ional accele a ion (m/s^2)
% Gene a e consis en andom hou selec ions o bo h analyses
andom_hou s = andi( o al_hou s, num_ epe i ions, 1);
%% 2. Calcula e As onomical Tide
Z0 = 33.37; % Mean sea le el in cm
% Ha monic componen s
ha monics = {
'SA', 0.000114, 7.62, 249.24;
'M2', 0.080511, 3.97, 207.67;
'K1', 0.041781, 3.7, 164.87;
'O1', 0.038731, 2.4, 103.02;
'S2', 0.083333, 1.35, 229.34;
'P1', 0.041553, 1.26, 160.36;
'N2', 0.078999, 0.86, 196.71;
'S1', 0.041667, 0.66, 261.22;
'M4', 0.161023, 0.5, 346.59;
'K2', 0.083561, 0.4, 223.79;
'Q1', 0.037219, 0.31, 53.58;
'MS4', 0.163845, 0.31, 51.29;
'MN4', 0.159511, 0.19, 304.58;
'NU2', 0.079202, 0.15, 199.07;
'M3', 0.120767, 0.15, 157.55;
'2N2', 0.077487, 0.14, 184.27;
'MU2', 0.077689, 0.14, 173.01;
'L2', 0.082024, 0.11, 216.42;
'T2', 0.083219, 0.1, 193.49;
'MK4', 0.164073, 0.09, 55.84;
'SK3', 0.125114, 0.08, 109.04;
'SN4', 0.162333, 0.05, 6.84;
};
98
[A(i), B(i), C(i)] = deal(1.03, -0.01, 1.56);
equencies = cell2ma (ha monics(:, 2));
ampli udes = cell2ma (ha monics(:, 3));
phases_ adians = deg2 ad(cell2ma (ha monics(:, 4)));
as onomical_ ide = Z0 + sum(ampli udes .* cos(2 * pi * equencies .*
andom_hou s' + phases_ adians), 1)';
as onomical_ ide = as onomical_ ide / 100; % Con e o me es
%% 3. Gene a e Me eo ological Residue
mu = 0.5; % Mean (cen e )
sigma = 0.06; % Reduced s anda d de ia ion o be e sp ead wi hin
bounds
% Gene a e using unca ed no mal dis ibu ion app oach
z = andn(num_ epe i ions*2, 1); % Gene a e ex a alues o unca ion
z = mu + sigma * z;
alid_idx = z >= 0.3 & z <= 0.7; % Find alues wi hin bounds
esidue_ alues = z( alid_idx); % Keep only alid alues
esidue_ alues = esidue_ alues(1:num_ epe i ions); % Take equi ed
numbe o samples
%% 4. Calcula e Wa e Heigh s and Seasonal Pa ame e s
% De ine mon h anges
mon h_ anges = [
1, 744; % Janua y (31 days)
745, 1416; % Feb ua y (28 days)
1417, 2160; % Ma ch (31 days)
2161, 2880; % Ap il (30 days)
2881, 3624; % May (31 days)
3625, 4344; % June (30 days)
4345, 5088; % July (31 days)
5089, 5832; % Augus (31 days)
5833, 6552; % Sep embe (30 days)
6553, 7296; % Oc obe (31 days)
7297, 8016; % No embe (30 days)
8017, 8760 % Decembe (31 days)
];
% Ini ialize a ays
mon hs = ze os(num_ epe i ions, 1);
A = ze os(num_ epe i ions, 1);
B = ze os(num_ epe i ions, 1);
C = ze os(num_ epe i ions, 1);
% Assign seasonal cons an s
o i = 1:num_ epe i ions
hou = andom_hou s(i);
o mon h = 1:12
i hou >= mon h_ anges(mon h, 1) && hou <= mon h_ anges(mon h,
2)
mon hs(i) = mon h;
% Assign cons an s based on season
swi ch ue
case ismembe (mon h, [12, 1, 2]) % Win e
[A(i), B(i), C(i)] = deal(0.79, 0.22, 1.04);
case ismembe (mon h, 3:5) % Sp ing
[A(i), B(i), C(i)] = deal(1.04, -0.06, 1.57);
case ismembe (mon h, 6:8) % Summe
[A(i), B(i), C(i)] = deal(0.4, 0.14, 1.15);
case ismembe (mon h, 9:11) % Au umn
99
Tp( alid_indices) = 4.32 * Hs_ inal( alid_indices).^0.43;
end
b eak;
end
end
end
% Calcula e ini ial and adjus ed Hs
P = and(num_ epe i ions, 1);
Hs_ini ial = B + A .* (-log(1 - P)).^(1 ./ C);
% Pa ame e s o Hs adjus men
be a = 0.44;
alpha = 3.1;
gamma = 0.76;
adjus _indices = Hs_ini ial >= 3;
% Ini ialize and calcula e adjus ed Hs
Hs_adjus ed = nan(num_ epe i ions, 1);
P_adjus ed = and(sum(adjus _indices), 1);
Hs_adjus ed(adjus _indices) = alpha + be a .* (-log(1 - P_adjus ed)).^(1
./ gamma);
% Coun he numbe o alid (non-NaN) Hs_adjus ed alues
num_ alid_Hs_adjus ed = sum(~isnan(Hs_adjus ed));
pe cen ile_ empo al=(num_ alid_Hs_adjus ed/num_ epe i ions)*100;
%% 5. Wa e Di ec ion Analysis
wa e_di ec ions = {'N', 'NE', 'E', 'SE', 'S', 'SW', 'W', 'NW'};
di ec ion_p obabili ies = [4, 5, 25, 15, 24, 9, 9, 9];
% Calcula e cumula i e p obabili ies
cumula i e_p obs = cumsum(di ec ion_p obabili ies) /
sum(di ec ion_p obabili ies);
p ob_edges = [0; cumula i e_p obs(:)];
% Gene a e and assign di ec ions
P_di ec ion = and(sum(adjus _indices), 1);
di ec ion_indices = disc e ize(P_di ec ion, p ob_edges);
% Con e indices o di ec ions
wa e_di ec ions_a ay = cell(num_ epe i ions, 1);
wa e_di ec ions_a ay(adjus _indices) =
wa e_di ec ions(di ec ion_indices);
% Fil e alid di ec ions
wa e_di s_ca = ca ego ical(wa e_di ec ions_a ay(adjus _indices));
alid_di s_ca = ca ego ical( alid_di ec ions);
alid_di ec ion_mask = ismembe (wa e_di s_ca , alid_di s_ca );
alid_di ec ion_indices = alse(num_ epe i ions, 1);
alid_di ec ion_indices(adjus _indices) = alid_di ec ion_mask;
% C ea e inal Hs
Hs_ inal = nan(num_ epe i ions, 1);
Hs_ inal( alid_di ec ion_indices) = Hs_adjus ed( alid_di ec ion_indices);
%% 6. Calcula e Wa e Pa ame e s
% Calcula e Tp
Tp = nan(num_ epe i ions, 1);
alid_indices = alid_di ec ion_indices & ~isnan(Hs_ inal);
100
% Calcula e deep wa e wa eleng h (L0)
L0 = nan(num_ epe i ions, 1);
L0( alid_indices) = (g * Tp( alid_indices).^2) ./ (2 * pi);
% Calcula e I iba en numbe (xi)
xi = nan(num_ epe i ions, 1);
xi( alid_indices) = slope ./ sq (Hs_ inal( alid_indices) ./
L0( alid_indices));
%% 7. Calcula e Wa e Run-up (R2%)
R2 = nan(num_ epe i ions, 1);
% Fo xi < 3
low_xi_indices = alid_indices & (xi < 3);
R2(low_xi_indices) = 0.73 * slope * sq (Hs_ inal(low_xi_indices) .*
L0(low_xi_indices));
% Fo xi >= 3
high_xi_indices = alid_indices & (xi >= 3);
R2(high_xi_indices) = 1.1 * (0.35 * slope * sq (Hs_ inal(high_xi_indices)
.* L0(high_xi_indices)) + ...
sq (Hs_ inal(high_xi_indices) .* L0(high_xi_indices) .* (0.563 *
slope^2 + 0.004)) / 2);
o al_wa e _le el = as onomical_ ide + esidue_ alues + R2;
%% 8. Sea Le el Rise (SLR) + Subcidence Scena io
% Sea Le el Rise Scena ios (in me es)
% Mode a e scena io (RCP4.5)
sl _2050_mode a e = 0.25; % 25 cm ise by 2050
sl _2100_mode a e = 0.50; % 50 cm ise by 2100
% High emission scena io (RCP8.5)
sl _2050_high = 0.30; % 30 cm ise by 2050
sl _2100_high = 0.80; % 80 cm ise by 2100
% Local subsidence a e (me es/yea )
subsidence_ a e = 0.003; % 3 mm/yea
% Time pe iods o analysis
yea s = [2025, 2050, 2100];
scena ios = {'Cu en ', 'RCP4.5', 'RCP8.5'};
% Calcula e SLR o di e en scena ios and yea s
sl _scena ios = ze os(leng h(yea s), leng h(scena ios));
o i = 1:leng h(yea s)
yea = yea s(i);
yea s_ om_now = yea - 2025;
% Cu en scena io (only subsidence)
sl _scena ios(i, 1) = subsidence_ a e * yea s_ om_now;
% RCP4.5 (mode a e) scena io
i yea <= 2050
sl _scena ios(i, 2) = (sl _2050_mode a e * (yea - 2025)/(2050 -
2025)) + ...
(subsidence_ a e * yea s_ om_now);
101
else
sl _scena ios(i, 2) = sl _2050_mode a e + ...
(sl _2100_mode a e - sl _2050_mode a e) *
(yea - 2050)/(2100 - 2050) + ...
(subsidence_ a e * yea s_ om_now);
end
% RCP8.5 (high) scena io
i yea <= 2050
sl _scena ios(i, 3) = (sl _2050_high * (yea - 2025)/(2050 -
2025)) + ...
(subsidence_ a e * yea s_ om_now);
else
sl _scena ios(i, 3) = sl _2050_high + ...
(sl _2100_high - sl _2050_high) * (yea -
2050)/(2100 - 2050) + ...
(subsidence_ a e * yea s_ om_now);
end
end
%% 9. Failu e, Realiabili y Index and Con e gence wi h espec o Num. o
Repe i ions
%% 9.1. Highe C i ical heigh
%% 9.1.1. Analysis o Highe C i ical Heigh
ole ance = 0.0001; % Con e gence ole ance
window_size = 200; % Window size o s abili y
s ep_size = 50; % S ep size o sampling
max_samples = num_ epe i ions;
sample_sizes = s ep_size:s ep_size:max_samples;
% Ini ialize a ays
con e gence_ o al_high = ze os(leng h(sample_sizes), leng h(yea s),
leng h(scena ios));
con e gence_poin s_high = ze os(leng h(yea s), leng h(scena ios));
ailu e_p ob_ o al_high = ze os(leng h(yea s), leng h(scena ios));
eliabili y_indices_high = ze os(leng h(yea s), leng h(scena ios));
% Calcula e base ailu e p obabili y and eliabili y index o high
c i ical heigh
base_ ailu e_high = sum( o al_wa e _le el >= c i ical_heigh _high) /
num_ epe i ions;
ma gin_sa e y_high = c i ical_heigh _high - o al_wa e _le el;
be a_base_high = mean(ma gin_sa e y_high, 'omi nan') /
s d(ma gin_sa e y_high, 'omi nan');
% Calcula e con e gence, ailu e p obabili ies, and eliabili y indices
o i = 1:leng h(yea s)
o j = 1:leng h(scena ios)
adjus ed_wa e _le el = o al_wa e _le el + sl _scena ios(i,j);
% Failu e p obabili y
ailu es = sum(adjus ed_wa e _le el >= c i ical_heigh _high);
ailu e_p ob_ o al_high(i,j) = ailu es / num_ epe i ions;
% Reliabili y index
ma gin_sa e y = c i ical_heigh _high - adjus ed_wa e _le el;
mean_ma gin = mean(ma gin_sa e y, 'omi nan');
s d_ma gin = s d(ma gin_sa e y, 'omi nan');
eliabili y_indices_high(i,j) = mean_ma gin / s d_ma gin;

102
% Con e gence
o k = 1:leng h(sample_sizes)
n = sample_sizes(k);
ailu es_n = sum(adjus ed_wa e _le el(1:n) >=
c i ical_heigh _high);
con e gence_ o al_high(k,i,j) = ailu es_n / n;
end
% Find con e gence poin
o k = window_size:leng h(sample_sizes)
window = con e gence_ o al_high((k-window_size+1):k,i,j);
i s d(window) < ole ance
con e gence_poin s_high(i,j) = sample_sizes(k-
window_size+1);
b eak;
end
end
end
end
% Sa e essen ial a iables o heigh calcula ions
sa e('base_mon e_ca lo.ma ', ...
' andom_hou s', % Fo consis en ime sampling
'as onomical_ ide', % Base as onomical ide calcula ions
' esidue_ alues', % Me eo ological esiduals
'Hs_ini ial', % Ini ial wa e heigh s
'Hs_adjus ed', % Wa e heigh s a e 3m h eshold
'adjus _indices', % Indices o adjus ed heigh s
'P_di ec ion', % Random alues o di ec ion assignmen
'mon hs', % Mon h assignmen o seasonali y
'P', % O iginal andom alues o ini ial Hs
'P_adjus ed'); % Random alues o adjus ed Hs
103
2. Ma lab used unc ions
1. `clea all`
- Remo es all a iables om he wo kspace memo y.
- Cleans up he en i onmen be o e unning new code.
2. `close all`
- Closes all igu e windows.
- Used o ensu e a clean isualiza ion en i onmen .
3. `load()`
- Loads da a om a MAT- ile in o he wo kspace.
- Used o load 'my_ abucado .ma ' in his sc ip .
4. ` andi()`
- Gene a es andom in ege s om a uni o m dis ibu ion.
- Used o gene a e andom hou selec ions wi hin o al_hou s ange.
5. `cell2ma ()`
- Con e s a cell a ay in o a egula a ay.
- Used o con e ha monic componen s da a in o nume ical a ays.
6. `deg2 ad()`
- Con e s angles om deg ees o adians.
- Used o con e phase angles o as onomical ide calcula ions.
7. ` andn()`
- Gene a es no mally dis ibu ed andom numbe s.
- Used in me eo ological esidue calcula ion.
8. ` and()`
- Gene a es uni o mly dis ibu ed andom numbe s.
- Used o p obabili y calcula ions in wa e heigh analysis.
9. `deal()`
- Assigns inpu alues o ou pu a iables.
- Used o assign seasonal cons an s (A, B, C).
10. `ismembe ()`
- Re u ns a ay elemen s ha a e membe s o a se .
- Used o checking alid wa e di ec ions and mon hs.
11. `sum()`
- Calcula es he sum o a ay elemen s.
- Used in a ious p obabili y and coun ing calcula ions.
104
12. `cumsum()`
- Calcula es cumula i e sum o a ay elemen s.
- Used in wa e di ec ion p obabili y calcula ions.
13. `disc e ize()`
- Bins nume ic da a in o ca ego ical da a.
- Used in wa e di ec ion analysis.
14. `ca ego ical()`
- Con e s da a o ca ego ical a ay ype.
- Used o wa e di ec ion ca ego iza ion.
15. `sq ()`
- Calcula es squa e oo .
- Used in a ious wa e pa ame e calcula ions.
16. `ze os()`
- C ea es a ay o ze os.
- Used o ini ialize a ious a ays h oughou he sc ip .
17. `nan()`
- C ea es a ay o NaN (No a Numbe ) alues.
- Used o ini ialize a ays o wa e pa ame e s.
18. `mean()`
- Calcula es a e age o a ay elemen s.
- Used wi h 'omi nan' pa ame e o eliabili y calcula ions.
19. `s d()`
- Calcula es s anda d de ia ion.
- Used wi h 'omi nan' pa ame e o eliabili y calcula ions.
20. ` p in ()`
- Ou pu s o ma ed ex o display esul s.
- Though no explici ly shown in he p o ided code snippe , i is commonly used o
displaying esul s in MATLAB.
21. ` loo ()`
- Rounds numbe s down o he nea es in ege .
- Though no explici ly used in he p o ided code, i would be used o in e al
calcula ions.
22. `cell()`
- C ea es cell a ays.
- Used in he sc ip o handling he wa e di ec ions a ay
- Example in code: `wa e_di ec ions = {'N', 'NE', 'E', 'SE', 'S', 'SW', 'W', 'NW'};`.
105
23. `leng h()`
- Re u ns he leng h o a ays o ec o s.
- Used ex ensi ely in he code o loop i e a ions and a ay sizing.
- Example in code: ` o i = 1:leng h(yea s)`.