Discontinuous stochastic forcing in Greenland ice core data

ORIGINAL ARTICLE
Clima e Dynamics (2025) 63:465
h ps://doi.o g/10.1007/s00382-025-07880-9
Keno Rieche s, And eas Mo and Leona do Rydin Go jão ha e
con ibu ed equally o his wo k.
And eas Mo
[email p o ec ed]
1 Clima e Physics, Max Planck Ins i u e o Me eo ology,
Hambu g, Ge many
2 Ea h Sys em Modelling, School o Enginee ing and Design,
Technical Uni e si y o Munich, Munich, Ge many
3 Resea ch Domain IV – Complexi y Science, Po sdam
Ins i u e o Clima e Impac Resea ch, 14473 Po sdam,
Ge many
4 Depa men o Epilep ology, Uni e si y o Bonn Medical
Cen e , 53105 Bonn, Ge many
5 Helmhol z Ins i u e o Radia ion and Nuclea Physics,
Uni e si y o Bonn, 53115 Bonn, Ge many
6 In e disciplina y Cen e o Complex Sys ems, Uni e si y o
Bonn, 53175 Bonn, Ge many
7 Depa men o Compu e Science, OsloMe – Oslo
Me opoli an Uni e si y, 0130 Oslo, No way
8 School o Economics, Inno a ion and Technology, K is iania
Uni e si y o Applied Sciences, Ki kega a 24-26, 0153 Oslo,
No way
9 Global Sys ems Ins i u e and Depa men o Ma hema ics,
Uni e si y o Exe e , Exe e , UK
10 Ins i u e o Ene gy and Clima e Resea ch (IEK-10),
Fo schungszen um Jülich, 52428 Jülich, Ge many
11 Ins i u e o Theo e ical Physics, Uni e si y o Cologne,
50937, Köln, Ge many
12 Depa men o En i onmen al Sciences Facul y o Science,
Open Uni e si y, Hee len, The Ne he lands
13 Facul y o Science and Technology, No wegian Uni e si y o
Li e Sciences, 1432 Ås, No way
Abs ac
Paleoclima e p oxy eco ds om G eenland ice co es, a chi ing e.g.
δ18
O as a p oxy o su ace empe a u e, show ha
sudden clima ic shi s called Dansgaa d–Oeschge e en s (DO) occu ed epea edly du ing he las glacial in e al. They
comp ised subs an ial wa ming o he A c ic egion om cold o milde condi ions. Concomi an ab up changes in he
dus concen a ions o he same ice co es sugges ha sudden eo ganisa ions o he hemisphe ic-scale a mosphe ic ci cula-
ion ha e accompanied he wa ming e en s. Genuine bis abili y o he No h A lan ic clima e sys em is commonly hypo h-
esised o explain he exis ence o s adial (cold) and in e s adial (milde ) pe iods in G eenland. Howe e , he physical
mechanisms ha d o e ab up ansi ions om he s adial o he in e s adial s a e, and mo e g adual ye s ill ab up e e se
ansi ions, emain deba ed. He e, we conduc a one-dimensional da a-d i en analysis o he G eenland empe a u e and
a mosphe ic ci cula ion p oxies unde he pu iew o s ochas ic p ocesses. We ake he K ame s–Moyal equa ion o es i-
ma e each p oxy’s d i and di usion e ms wi hin a Ma ko ian model amewo k. We hen assess noise con ibu ions
beyond Gaussian whi e noise. The esul ing s ochas ic di e en ial equa ion (SDE) models ea u e a monos able d i o
he G eenland empe a u e p oxy and a bis able one o he a mosphe ic ci cula ion p oxy. Indica o s o discon inui y in
s ochas ic p ocesses sugges o include highe -o de e ms o he K ame s–Moyal equa ion when modelling he G eenland
empe a u e p oxy’s e olu ion. This cons i u es a quali a i e di e ence in he cha ac e is ics o he wo ime se ies, which
should be u he in es iga ed om he s andpoin o clima e dynamics.
Recei ed: 22 Feb ua y 2025 / Accep ed: 15 Sep embe 2025
© The Au ho (s) 2025
Discon inuous s ochas ic o cing in G eenland ice co e da a
KenoRieche s1· And easMo 2,3 · KlausLehne z4,5,6 · Ped o G.Lind7,8 · NiklasBoe s2,3,9 ·
Di kWi hau 10,11 · Leona do RydinGo jão12,13
1 3
K. Rieche s e al.
1 In oduc ion
Paleoclima e p oxy eco ds p o ide e idence o pas
ab up clima e shi s om egional o a leas hemisphe ic
scale (e.g. Men iel e al. 2020; B o kin e al. 2021; Boe s
e al. 2022). Long- e m clima e simula ions sugges ha
an h opogenic global wa ming could igge s uc u ally
simila ansi ions in se e al Ea h sys em componen s in
he u u e, i.e., ha hese componen s could ‘ ip’ o a quali-
a i ely di e en s a e (e.g. Len on e al. 2008, 2019; Boe s
2021; A ms ong e al. 2022; Boul on e al. 2022; Wang e al.
2023). Such ca as ophic shi s would ha e se e e conse-
quences on socie ies and ecosys ems and may e en unleash
eedbacks, u he inc easing he global mean empe a u e.
Howe e , he assessmen o po en ially upcoming ipping
poin s is challenging as he capabili y o mode n complex
clima e models o simula e clima e ipping dynamics is s ill
limi ed (Valdes 2011; Liu e al. 2017; Wang e al. 2023). In
ligh o his, he s udy o pas ab up clima e shi s may p o-
ide insigh s in o he p ocesses in ol ed in clima e ipping
e en s. Fu he mo e, pas e en s may se e as benchma ks
o he pe o mance o ully coupled models in simula ing
he non-linea and high-dimensional dynamics ha could
lead o ipping e en s. In his con ex , we eassess he e wo
p oxy ime se ies om he NGRIP ice co e (No h G een-
land Ice Co e P ojec s membe s 2004), which ea u e p o-
nounced imp in s o ab up clima ic ansi ions, by means o
he K ame s–Moyal equa ion.
Agnos ic ime se ies models, i.e., models whose dynam-
ics appea o ep oduce na u e bu a e no en i ely based on
physical mechanisms, ha e played a majo ole in u he -
ing he deba e on clima e ipping phenomena (e.g. Rieche s
e al. 2023a; Boe s e al. 2017; Mi sui and C uci ix 2017;
Kwasniok 2013; Lohmann and Di le sen 2018; Dakos e al.
2008; Bochow and Boe s 2023). The abili y o p oduce
quan i a i ely simila dynamical beha iou building only
on heu is ic physical assump ions acili a es he s a is ical
analysis o ipping phenomena, employing me hods o s o-
chas ic analysis (Len on e al. 2012; Mo and Boe s 2024;
Mo e al. 2024). The common concep o a clima e ipping
elemen is ha o a dynamical sys em whose cu en s able
equilib ium s a e is p one o annihila ion in a dynamic bi u -
ca ion (Sche e e al. 2009; Ashwin e al. 2012; Boe s e al.
2022). This ypically in ol es he educ ion o complex,
high-dimensional dynamics o jus a ew (i no one) sum-
ma y obse ables ha may be modelled in e ms o s ochas-
ic di e en ial equa ions (SDEs), i.e., as andom dynamical
sys ems. The ein, he noise e m e lec s he ac ion o he
un esol ed dynamics on he summa y obse able (Has-
selmann 1976). A common choice is o o ce he esol ed
a iables wi h Gaussian whi e noise, bu his app oach may
be o e ly simplis ic in many si ua ions. In pa icula , in he
con ex o clima e ipping poin s, a de ia ion om Gauss-
ian whi e noise has impo an implica ions o he de ec ion
o ea ly wa ning signals and o he p obabili y o p ema-
u e noise-induced ipping (Di le sen 1999; Luca ini e al.
2022; Benson e al. 2024; Kuehn ey al. 2022; Mo and
Boe s 2024).
He e, we in es iga e he amous hea y-oxygen
δ18
O eco d om he NGRIP ice co e (No h G eenland Ice
Co e P ojec s membe s 2004). The da a shows ha epea ed
decadal-scale wa ming e en s o egionally up o 16
◦
C in
ampli ude, known as Dansgaa d–Oeschge e en s, punc-
ua ed he No h A lan ic clima e h oughou he las gla-
cial in e al (Dansgaa d e al. 1984; B oecke e al. 1985;
Johnsen e al. 1992; Dansgaa d e al. 1993; Kindle e al.
2014). The sudden empe a u e inc eases we e ollowed by
a phase o mode a e cooling be o e he empe a u es ul i-
ma ely elaxed back o colde le els in a second phase o
mo e ab up cooling. The wo dis inc cold and mild egimes
a e e med s adials and in e s adials, espec i ely.
In line wi h he SDE app oach ou lined abo e, we ega d
he
δ18
O and dus concen a ion eco ds as ealisa ions o
one-dimensional Ma ko p ocesses and es ima e he co -
esponding KM coe icien s (Taba 2019). The wo eco ds
exhibi concomi an shi s, which a e in e p e ed as sudden
adjus men s o global mean empe a u e and eo ganisa-
ions o he a mosphe ic ci cula ion o a leas hemisphe ic
scale (Fuh e e al. 1999; Ru h e al. 2003, 2007; Schüp-
bach e al. 2018). P e ious s udies ha e mo i a ed h ough
s a is ical means he employmen o a Ma ko ian ame-
wo k o hese dynamics Rieche s e al. (2023b); Kwas-
niok (2013). The e ha e also been concep ual a gumen s
o ime-scale sepa a ion ha lend his amewo k c edence
Go wald (2021); Rieche s e al. (2024). Unde his model-
ling assump ion, he KM coe icien s a e closely ela ed o
he Fokke –Planck equa ion o ime-e ol ing di usi e sys-
ems. We es ima e om he da a and subsequen ly compa e
he wo K ame s–Moyal expansions wi h espec o hei
implied s ochas ic model s uc u e. Speci ically, we in es-
iga e whe he he dynamics can each be ep esen ed by a
canonical Lange in app oach o whe he a discon inuous
noise componen , such as Poisson jump di usion, is needed.
This a icle is s uc u ed as ollows: In Sec. 2 we b ie ly
in oduce he wo paleo-clima ic p oxies ha we examine.
Subsequen ly, in Sec. 3, we de ail he K ame s–Moyal
expansion in one dimension as he p ime me hod o con-
s uc ime se ies models including noise and possibly dis-
con inuous elemen s. Sec ion 4 p esen s he esul s o his
analysis: He ein, we show he mono- and bis abili y o he
ob ained models o he wo eco ds and discuss he need o
choose a noise model di e en om Gaussian whi e noise.
1 3
465 Page 2 o 14
Discon inuous s ochas ic o cing in G eenland ice co e da a
In Sec. 5 we discuss ou indings and ela e hem o p e i-
ous wo k. Sec. 6 summa ises ou key indings and d aws
conclusions.
2 Da a and p e-p ocessing
This wo k elies on he
δ18
O and dus concen a ion
eco ds ob ained by he No h G eenland Ice Co e P ojec
(NGRIP) (Ru h e al. 2003; No h G eenland Ice Co e P oj-
ec s membe s 2004; Gkinis e al. 2014). F om 1404.75 m
o 2426.00 m o dep h he join eco d is p o ided a 5 cm
equidis an esolu ion. This ansla es o he ime span om
59945 y o 10276 y b2k (be o e 2000 CE) wi h
∼5
y es-
olu ion o he oldes and sub-annual esolu ion o he mos
ecen pa o he eco d (Fig. 1a and b). Fo he analysis,
he da a was escaled, binned o an equidis an ime axis o
5-yea esolu ion, de ended, and no malised (see Appendix
A o de ails).
The concen a ion o dus , i.e., he numbe o pa icles
wi h a diame e la ge han 1
µ
m pe ml, is commonly
in e p e ed as a p oxy o he s a e o he hemisphe ic a mo-
sphe ic ci cula ion (e.g. Fische e al. 2007; Ru h e al. 2007;
Schüpbach e al. 2018; E ha d e al. 2019). In pa icula , he
dus s o m ac i i y and d yness o e Eas Asian desse s, he
s eng h and posi ion o he pola je , and local p ecipi a-
ion pa e ns go e n he emission, anspo , and deposi ion
o he dus , espec i ely (Fische e al. 2007; E ha d e al.
2019). Co espondingly, he subs an ial changes in he dus
concen a ions a DO e en s a e in e p e ed as la ge-scale
eo ganisa ions o he No he n Hemisphe e’s a mosphe ic
ci cula ion. In ag eemen wi h a widesp ead con en ion,
we escale he dus eco d by aking he ne nega i e loga-
i hm (e.g. Di le sen 1999; Mi sui and C uci ix 2017; Boe s
e al. 2017; Rieche s e al. 2023a). In his o m, he dus
eco d exhibi s a high deg ee o co ela ion wi h he
δ18
O
eco d (Boe s e al. 2017).
In o de o educe he in luence o slow changes in he
backg ound clima e, we es ic ed he analysis o he pe iod
59–27 ky b2k and applied u he de ending wi h espec
o a No he n Hemisphe e empe a u e econs uc ion p o-
ided by Snyde (2016) (see Fig. 1c and d and App. Appen-
dix A). The concen a ion o s able wa e iso opes exp essed
as
δ18
O alues in uni s o pe mil is a p oxy o he si e em-
pe a u e a he ime o p ecipi a ion (Jouzel e al. 1997; Gki-
nis e al. 2014).
Fig. 1 T ajec o ies o he 20-yea mean o
δ18
O (a) and acco dingly
esampled dus concen a ions (b) om he NGRIP ice co e in G een-
land, om 122 ky and 107 ky o 10 ky be o e 2000 CE (b2k),
espec i ely (Ru h e al. 2003; Rasmussen e al. 2014; Seie s ad e al.
2014). The dus da a is gi en as he nega i e na u al loga i hm o he
ac ual dus concen a ions, in o de o acili a e compa ison o he
δ18
O da a. Panels (c) and (d) show he same p oxies bu a a highe eso-
lu ion o 5 yea s (No h G eenland Ice Co e P ojec s membe s 2004;
Gkinis e al. 2014; Ru h e al. 2003) and o e he sho e pe iod om
59 o 27 ky b2k. The analysis p esen ed in his s udy was cons ained
o his segmen o he eco ds. The wo p oxy ime se ies in (c) and
(d) ha e been de ended by linea ly eg essing he da a agains econ-
s uc ed global mean su ace empe a u es (Snyde 2016) and emo -
ing he appa en backg ound- empe a u e-d i en slow change. The
g ey shadings ma k he G eenland in e s adial (GI) in e als acco ding
o (Rasmussen e al. 2014). All da a a e shown on he GICC05 ch o-
nology (Vin he e al. 2006; Rasmussen e al. 2006; Ande sen 2006;
S ensson e al. 2008). The da a we e binned o equidis an ime esolu-
ion om i s o iginal 5 cm dep h esolu ion (see App. Appendix A o
u he de ails on he da a p ocessing (Rieche s e al. 2023a))
1 3
Page 3 o 14 465
K. Rieche s e al.
1974), a Ma ko s ochas ic p ocess o he o m Eq. (1)
gene ally ea u es discon inuous pa hs wi h non-ze o p ob-
abili y. Pa h-wise con inui y is only one o many no ions
o con inui y in s ochas ic p ocesses. Ano he is he con i-
nui y c i e ion o Ma ko p ocesses p o ided by Ga dine
(2009), which equi es o a p ocess o be con inuous ha
C
(x, , δ) = lim
τ→0
1
τP(|x +τ−x |>δ)
= lim
τ→0
1
τˆ
|
x
′−
x
|
>δ
p(x′, +τ|x, )dx′!
=0,
(5)
o all
δ
, x, and . In wo ds, his means ha he p obabili y
o a pa icle de ia ing om a e e ence posi ion mo e han
δ
in a ime in e al
τ
dec eases as e han linea ly wi h
τ
.
The p esence o highe -o de KM coe icien s in he co -
esponding K ame s–Moyal equa ion is a necessa y, ye no
su icien c i e ion o a gi en p ocess o be discon inuous
unde his la e no ion.
3.1 Es ima ing K ame s–Moyal coe icien s
The cen al en y poin o his wo k is Eq. (3). I p o ides a
means o es ima e he KM coe icien s
Dm(x)
di ec ly om
da a, i.e., om a eco ded ealisa ion o a s ochas ic p ocess,
p o ided ha he ollowing assump ions a e ul illed ( o a
easonable deg ee):
i) The obse ed p ocess is a Ma ko p ocess,
ii) he p ocess is ime-homogeneous, i.e., he dynamics did
no change o e ime,
iii) he s a e space is sampled su icien ly densely,
i ) and he sampling ime is sho compa ed o he cha ac-
e is ic ime scale o he dynamics.
Unde hese condi ions, he e alua ion o he condi ional
s a is ical momen s
M(x, τ)
a he sho es a ailable ime
lag
∆
gi en by he sampling a e yields a good es ima e o
he KM coe icien s:
ˆ
D
m(x)=
1
m!
1
∆ ⟨
(x +∆
−
x )m
|
x =x
⟩≈
Dm(x)
,
(6)
whe ein he ensemble a e age in Eq. (3) is eplaced by he
a e age o e he a ailable da a
⟨·⟩
. Ou nume ical imple-
men a ion o Eq. (6) is based on he Nada aya–Wa son es i-
ma o which is de ailed in App. Appendix B.
3 Me hods
Ou s a ing poin is a ( ime-homogeneous) Ma ko s o-
chas ic p ocess
x
o he o m
dx = (x )d +σ(x )dξ ,
(1)
whe e
dξ
deno es an a bi a y unco ela ed s ochas ic
o ce. The empo al e olu ion o he associa ed condi ional
p obabili y unc ion
p(x, +τ|x′
, )
hen ollows he K am-
e s–Moyal equa ion (K ame s 1940; Moyal 1949; Kampen
1961; Ga dine 2009; Risken and F ank 1996; Taba 2019):
∂
∂τ p(x, +τ|x′
, )=
∞
∑
m=1
(
−∂
∂x
)m
Dm(x)p(x, +τ|x′
, )
.
(2)
The K ame s–Moyal (KM) coe icien s
Dm(x)
a e ela ed
o he condi ional momen s
Mm(x, τ)
o o de m o he s o-
chas ic a iable x a a ime-lag
τ
by
D
m(x)=
1
m!lim
τ→0
1
τMm(x, τ)
=1
m!
lim
τ→0
1
τˆ
(x′
−
x )mp(x′, +τ
|
x, )dx′
.
(3)
In he special case ha he s ochas ic o ce in Eq. (1) is gi en
by Gaussian whi e noise (i.e., i can be exp essed by he
inc emen s o a Wiene p ocess
W
), only he i s wo e ms
on he igh o Eq. (2) con ibu e and he K ame s–Moyal
equa ion educes o he be e -known Fokke –Planck equa-
ion (Fokke 1913, 1914; Planck 1917). Wi h
dξ =dW
,
Eq. (1) becomes he Lange in equa ion and he esul ing
p ocess is hen e e ed o as a Lange in p ocess1. Fo Lan-
ge in p ocesses he ela ion
D
1(x)= (x)and D2(x)=
1
2
σ2(x)
,
(4)
be ween he KM coe icien s, he d i (x) and he di usion
σ(x)
, holds in gene al.
The o he way a ound, i highe -o de momen s con ib-
u e o he K ame s–Moyal equa ion, he unde lying p o-
cess canno be a s anda d Lange in p ocess. In ha case,
ξ
does no co espond o a Wiene p ocess bu has ins ead a
mo e complex o m. Howe e , he i s wo KM coe icien s
would s ill be domina ed by he p ocess’ d i and di usion.
While a Lange in p ocess consis s, wi h p obabili y 1,
o con inuous sample pa hs (e.g. Theo em 5.1.1 in A nold
1 The e is no ag eemen on he use o he e m Lange in p ocess.
Some au ho s conside Lé y-d i en equa ions as such Lange in equa-
ions, o he s p e e o e e o Lange in p ocesses as hose ha a e
solely d i en by Gaussian/B ownian noise.
1 3
465 Page 4 o 14
Discon inuous s ochas ic o cing in G eenland ice co e da a
Lange in om jump-di usion p ocesses, namely he
Θ
- a io
Θ(
x, τ)=
3
M2
(
x, τ
)2
M4(x, τ)∼{1
,
Lange in
,
1
τ
,jump-diffusion, (9)
and he Q- a io (Lehne z e al. 2018)
Q
(x, τ)=
M
6
(x, τ)
5
M
4(
x, τ
)∼{
τ,
Lange in
,
cons an ,jump-diffusion. (10)
Fo de ails on he de i a ion o hese ela ionships, we e e
he in e es ed eade o (Taba 2019). Obse ing ei he o
he scalings gi en in Eqs. (9) and (10), espec i ely, can aid
in deciding be ween employing a Lange in o jump-di u-
sion model.
These ela ionships a e speci ically de i ed o he jump-
di usion model. Compa ed o he a io o
D4
and
D2
dis-
cussed abo e, he esul s he e a e mo e p one o in alidi y
due o unjus i ied modelling assump ions on he eal da a.
Fo di e en noise models han he ones in oduced abo e,
di e en scaling beha iou s o hese a ios wi h espec o
τ
will a ise. Da a om, e.g., a non-Ma ko ian sys em may, on
he o he hand, exhibi he desc ibed beha iou while ac u-
ally ha bou ing en i ely di e en in e nal dynamics. In his
wo k, we ocus on dis inguishing be ween he Lange in and
Poisson jump-di usion models as wo a che ypical (dis-)
con inuous s ochas ic models. Obse ing any o he scaling
in Q o
Θ
may hin a a hi d model being mo e app op i-
a e o ep oduce he ime se ies dynamics. Howe e , in he
con ex o con inuous e sus discon inuous s ochas ic mod-
els, conside ing he wo discussed models yields essen ial
in o ma ion.
4 Resul s
Figu e 2 shows he i s and second KM coe icien s, and
he a io o he second o he ou h KM coe icien s, as
es ima ed om he dus and
δ18
O ime se ies acco ding o
Eq. (6). The co esponding
Θ
and Q a ios a e p esen ed in
Fig. 3.
a Dus eco d: Fo he dus , he cons uc ed d i
D1(x)
in Fig. 2b exhibi s wo sepa a e s able s a es ha ma ch he
maxima o he p obabili y densi y unc ion in Fig. 2a. The
second KM coe icien
D2(x)
in Fig. 2c is app oxima ely
cons an . The a io be ween he ou h and he second KM
coe icien s in Fig. 2d is smalle han 0.1 on he en i e s a e
space p obed by he ime se ies. Fo la ge po ions o he
dus ’s s a e space, we ind in Fig. 3 a dec ease o he
Θ(x, τ)
a io wi h inc easing
τ
, simila o a
1/τ
beha iou . This
applies, in pa icula , a he s able equilib ia o he d i ,
3.2 Es ima o s o discon inuous mo ion
Once he KM coe icien s a e es ima ed om he da a, one
can d aw in e ence on he mos i ing choice o he noise
model
dξ
. Vanishing highe -o de momen s (
m>2
) clas-
si y he model as a Lange in p ocess. In con as , demon-
s able con ibu ions o hese momen s sugges ha he
p ocess is bes modelled by including noise beyond a Wie-
ne p ocess (see e.g. Kampen 1961; Van Kampen 2007;
Ga dine 2009; Taba 2019; Lin 2023).
The ini e sampling ime s ep
∆
in oduces a bias o
he es ima o s
ˆ
Dm(
x
)
(Ku h e al. 2021). As a conse-
quence, e en o a Lange in p ocess he expec ed alues
o he highe -o de KM es ima o s di e om ze o. A i s
p agma ic me ic o disce n whe he a s udied p ocess is a
Lange in p ocess o no is o e alua e he a io be ween he
ou h KM coe icien and he second, i.e.,
D4(x)/D2(x)
.
This gauges he dis ibu ional ail o all immedia e dis u -
bances o igina ing om x. I he e o e o e s a non-pa ame -
ic insigh in o whe he a a ail o dis u bances is needed o
ec ea e he dynamics a he conside ed sampling a e. Such
conclusions would be la gely model-independen and do no
explici ly ely on he Ma ko iani y o he da a. Small alues
≲0.1
a e ypically ega ded as a jus i ica ion o a Lange in
desc ip ion. Values
D4(x)/D2(x)≳0.1
poin o non-di u-
si e mo ion (i.e., o cing beyond Gaussian whi e noise).
This me ic o e s a i s insigh in o whe he a discon inu-
ous noise e m
ξ
is needed o model he p ocess (Gao e al.
2016; Lu and Duan 2020; Luca ini e al. 2022).
When he Lange in p ocess model is con as ed wi h a
jump-di usion model o he o m (Taba 2019; Lin 2023)
d
x
=
(
x
)d
+
σ
(
x
)d
W
+
η
(
x
)d
J
(λ)
,
(7)
he assessmen can be u he e ined. He e,
J(
λ
)
deno es
a Poissonian jump p ocess cha ac e ised by he a e
λ
. The
jump ampli ude is de e mined by he Gaussian s ochas ic
a iable
η(x)
. Fo his speci ic p ocess model, he KM coe -
icien s ead (Taba 2019)
D
1
(x)= (x),
D2(x)=1
2σ(x)2+1
2λ(x)⟨η(x)2⟩,
D
m(x)= 1
m!
λ(x)
⟨
η(x)m
⟩
, o m>2
,
(8)
whe e
⟨·⟩
exp esses he expec ed alue.
Simila ly, he bias o he KM es ima o s de ined by
Eq. (6), when applied o a jump-di usion p ocess sampled
a ini e ime s ep
∆
, can be de i ed analy ically. These
conside a ions o e wo addi ional me ics o dis inguish
1 3
Page 5 o 14 465

K. Rieche s e al.
a iable. Finally, he a io
D4(x)/D2(x)≳0.3
is 10 imes
la ge o
δ18
O han o he dus . The
δ18
O eco d exhibi s
a mos ly cons an
Θ(x, τ)
- a io wi h espec o
τ
, as seen
in Fig. 3. I is sligh ly below bu s ill close o 1 o la ge
pa s o he s a e space. The co esponding
Q(x, τ)
- a io is
likewise cons an (
≈1
) wi h espec o
τ
, wi h a ia ions in
bo h di ec ions.
5 Discussion
The assessmen o he KM coe icien s and he scaling o
Θ
and Q a ios om he dus and he
δ18
O eco ds p o-
ides some insigh in o how o bes model he p oxy ime
se ies wi hin he amewo k o one-dimensional s ochas ic
p ocesses.
Fo he dus , we ind bis abili y o he es ima ed model’s
d i . The small
D4(x)/D2(x)
a io and he linea inc ease
o he
Q(x, τ)
wi h inc easing
τ
indica e ha his p ocess
can, in ac , be modelled as a Lange in p ocess. Only he
assessmen o he
Θ(x, τ)
a io calls his conclusion in o
ques ion. Fo a Lange in p ocess, his a io should be equal
o one, bu we obse e a
1/τ
-like scaling o small alues o
τ
, in line wi h an unde lying jump-di usion p ocess.
whe e he da a a ailabili y is he bes and ou es ima ion
is mos obus . The dus
Θ(x, τ)
- a io is close o 1 only in
a egion o i s s a e space whe e i s p obabili y densi y has
a local minimum (
−0.3≲dus ≲0.3
). The co espond-
ing
Q(x, τ)
- a io shows qui e a dis inc linea inc ease wi h
inc easing
τ
– a leas o small alues o
τ
. Fo la ge al-
ues o
τ
,
Q(x, τ)
is cons an .
b
δ18O
eco d: In he case o
δ18
O, he d i has only one
ze o-c ossing. This seems o explain he unimodal dis ibu-
ion o he da a, hough his b oade dis ibu ion could also
be caused by la ge obse a ional noise in he eco d. The
mono-s abili y o he d i would no be a ec ed by ime-
and s a e-independen obse a ional noise and can he e o e
be seen as a mo e di ec insigh in o he po en ial unde -
lying dynamics. We no e ha NGRIP da a p oduc s ha
p o ide
δ18
O concen a ions a a lowe ime esolu ion o
20- o 50-yea ime s eps exhibi a bimodal dis ibu ion.
Fo he pu poses o ou analysis, howe e , only he highes
a ailable sampling a e o ime se ies da a should be used
so as o cu ail he biases incu ed in he KM es ima ions.
Wi h espec o he no malised uni s, he i s and second
KM coe icien s o
δ18
O exceed hei coun e pa s o dus
by ac o s o app oxima ely 4 and 10, espec i ely. This
indica es ha
δ18
O was subjec ed o s onge noise while
simul aneously s onge de e minis ic o ces ac ed on he
Fig. 2 The p obabili y densi y unc ion (PDF) o (a) he dus and (e)
δ18
O. The non-pa ame ic es ima es o he (b, ) i s KM coe icien
D1(x)
and (c, g) he second KM coe icien
D2(x)
. The a io be ween
he ou h and he second KM coe icien
D4(x)/D2(x)
(d and h).
All KM coe icien s a e e alua ed a he sho es a ailable ime s ep
∆ =5
y o he ime se ies. The es ima ed dus d i is bis able, while
ha o
δ18
O is monos able. The second KM coe icien
D2(x)
is
ela i ely cons an o bo h eco ds. The a io
D4(x)/D2(x)
is small
(
≲0.1
) o he dus eco d. Ye , i is non-negligible o
δ18
O (
≳0.3
)
in la ge pa s o he s a e space, sugges ing ha he d i ing noise in a
s ochas ic model o hese ime se ies should no be exclusi ely Gauss-
ian whi e noise. De ails on he choice o ke nel and bandwid h used o
he KM coe icien es ima ion, as well as an analysis o he in luence o
he ke nel bandwid h, can be ound in App. B
1 3
465 Page 6 o 14
Discon inuous s ochas ic o cing in G eenland ice co e da a
wi h wo egimes in he p esence o a single equilib ium and
ime asymme y (Chechkin e al. 2003, 2004; Me zle and
Kla e 2004; Yang e al. 2020).
Gi en he high deg ee o isual simila i y be ween he
dus and he
δ18
O eco ds, he di e ences in he econ-
s uc ed po en ials and he a io be ween he ou h and he
second KM coe icien a e ema kable. This accen ua es
he need o ca e ul s a is ical analysis when de ising ime
se ies models o non-linea sys ems wi h ab up ansi ions.
Adop ing a gene alised Lange in equa ion wi h a bis able
d i e m, Di le sen (Di le sen 1999) showed ha he noise
in he calcium concen a ion eco d om he GRIP ice co e
can be modelled wi h an
α
-s able componen . Calcium con-
cen a ions a e ypically conside ed equi alen o dus con-
cen a ions (c . (Fuh e e al. 1993; Ru h e al. 2002, 2003;
Fische e al. 2007)). We canno di ec ly assess he p esence
o
α
-s able noise in he NGRIP dus eco d. This is because
noise models wi h in ini e s a is ical momen s, which can
We no e ha a simple Lange in model wi h a bis able
d i and pu ely di usi e noise can p oduce he egime
shi s obse ed in he dus eco d. Howe e , such a model is
unlikely o ep oduce he asymme ic shape o he in e s a-
dial phases e iden in he eco d.
Fo he
δ18
O eco d, he esul s a e exac ly he opposi e.
The cons uc ed d i unc ion exhibi s only a single s able
equilib ium. The obse ed quan i ies
D4(x)/D2(x)
and
Q(x, τ)
p o ide e idence o ele an con ibu ions om
highe -o de KM coe icien s. The
Θ(x, τ)
a io, howe e ,
is close o one in ag eemen wi h a Lange in model. A Lan-
ge in model oge he wi h he e idenced single equilib ium
o he d i unc ion clea ly ails o explain he wo egimes
o he
δ18
O eco d, and he appa en ime asymme y. Taken
oge he , we conclude ha he e idence speaks in a ou o
in oducing discon inui ies o he d i ing noise model a he
han agains i . Complex noise, i.e., noise beyond a Wie-
ne p ocess, could indeed be a way o ep oduce ime se ies
Fig. 3 The
Θ(x, τ)
- a io and
Q(x, τ)
- a io o he dus and he
δ18
O
concen a ion. (a) The
Θ(x, τ)
- a io o he dus is no close o 1 o
a la ge ange o
τ
and x, pa icula ly a ound he peaks o he bimodal
dus dis ibu ion. (b) F om
τ=5
o oughly
τ= 50
he
Q(x, τ)
- a io
inc eases linea ly wi h
τ
, consis en wi h a con inuous p ocess, ye
o
τ>50
he
Q(x, τ)
- a io is nea ly cons an , consis en wi h a dis-
con inuous p ocess. (c) On he one hand, he
Θ(x, τ)
- a io o he
δ18
O poin s o being di e en om 1 along he peak o he dis ibu ion,
ye no su icien ly conclusi e o asce ain i he
δ18
O is discon inu-
ous. (d) On he o he hand, he
Q(x, τ)
- a io is a guably cons an o e
τ
, consis en wi h a discon inuous
δ18
O. Fo isualisa ion pu poses,
Q(x, τ)
o
δ18
O is mul iplied by 0.6 o ma ch he scale o
Q(x, τ)
o he dus (Ha is e al. 2020; Vi anen e al. 2020; Rydin Go jão and
Mei inhos 2019; Rydin Go jão e al. 2023; Hun e 2007)
1 3
Page 7 o 14 465
K. Rieche s e al.
s abili y o he ac ual physical p ocesses. I is, howe e ,
no able ha hese indings a e inconsis en wi h he hypo h-
esis ha pas G eenland empe a u es we e go e ned by
in insically bis able dynamics (Li ina e al. 2010; Kwas-
niok 2013). Fo he a mosphe ic ci cula ion, he e is no such
inconsis ency. We s ess ha his inconsis ency wi h espec
o p e ious s udies a ises unde he applica ion o di e ing
modelling assump ions and da a p e-p ocessing p ocedu es.
The po en ial in luence o non-Ma ko iani y in he dynam-
ics o complex measu emen noise canno be quan i ied.
Disen angling hese con ounding e ec s is possible bu
demands la ge amoun s o da a Bö che e al. (2006). We
none heless main ain ha his no el pe spec i e is a alu-
able da a poin o u he concep ual and physical conside -
a ions o DO e en s.
We ound ha s ochas ic o cing should include e ms
beyond Gaussian whi e noise when modelling he
δ18
O
eco d. This ende s he Lange in app oach insu icien o
accu a ely ep oduce he ime se ies cha ac e is ics, d aw-
ing a en ion owa ds including discon inuous elemen s. Fo
he dus eco d, simila indica ions could be ound, hough
hese ha e no been as con incing.
In physical e ms, complex noise could ha e played a cen-
al ole in he eme gence o DO e en s. Ou analysis does
no p o ide di ec e idence o a causal ela ion be ween
discon inuous d i ing noise and he egime swi ches o he
No h A lan ic egion’s clima e du ing he las glacial. Ye , i
mo i a es u he explo a ion o his issue along he lines o
Go wald (2020) and Rieche s e al. (2023a). The possibili y
ha clima e ipping elemen s a e subjec o non-Gaussian
noise in oday’s clima e should ecei e g ea e conside -
a ion. The co esponding implica ions on he s abili y o
hese elemen s and he abili y o de ec ea ly wa ning sig-
nals should be in es iga ed.
Appendix A: Da a de ending
As men ioned in Sec. 2, his s udy ocuses on he pe iod
59–27 ky b2k. De ending o he da a is necessa y o
ensu e ha he ime se ies a e ime-homogeneous s a ion-
a y p ocesses, which is an unde lying assump ion o he
K ame s–Moyal analysis pe o med in ou in es iga ion. To
compensa e o he in luence o he backg ound clima e on
he clima e p oxy eco ds o dus and
δ18
O, we emo e a lin-
ea d i wi h espec o econs uc ed global a e age su ace
empe a u es (Snyde 2016) om bo h ime se ies. Figu e 4
illus a es he de ending scheme. Due o he wo- egime
na u e o he ime se ies, a simple linea eg ession would
o e es ima e he empe a u e dependencies. Ins ead, we
sepa a e he da a om G eenland s adials (GS) and G een-
land in e s adials (GI) and hen minimise he exp ession
be ound in
α
-s able dis ibu ions, a e inhe en ly incompa -
ible wi h he K ame s–Moyal amewo k. Ye , ou esul s
co obo a e he no ion ha G eenland ice co e eco ds bea
he signa u e o non-Gaussian noise, hough in ou analy-
sis his a ises p ima ily o he
δ18
O eco d. Rela ed o his,
Go wald (2020) ecen ly o mula ed a concep ual model o
DO e en s whe ein
α
-s able noise plays a cen al ole as
an e en igge , la e ex ended by Rieche s e al. (2023a).
F om he pe spec i e o heo e ical s ochas ic modelling, i
is wo h no ing ha he
α
-s able noise model leads o a pa h-
wise con inuous p ocess, in con as o he Poisson jump-
di usion model discussed in his wo k. Employing he
con inui y no ion o Eq. (5), howe e , bo h o hese models
would be conside ed discon inuous.
We ha e o s a e ha he in e p e a ion o highe -o de
KM coe icien s is no s aigh o wa d and depends on he
exac choice o he s ochas ic model. A di ec causal ela ion
be ween he DO e en s and discon inuous noise canno be
in e ed wi hou u he ado wi hin his s udy, bu he ole o
discon inui ies in he p oxy eco ds me i s u he in es iga-
ion. I has been obse ed in complex model simula ions ha
(s ochas ic) a mosphe ic anomalies can indeed d i e egime
changes in he No h A lan ic egion (D ij hou e al. 2013;
Kleppin e al. 2015). Toge he wi h he appa en ap i ude o
non-Gaussian noise models o G eenland empe a u e and
No he n Hemisphe e a mosphe ic ci cula ion p oxies, his
mo i a es u he esea ch on he e ec ha non-Gaussian
noise could ha e on clima e ipping elemen s in p esen -day
clima e.
I bo h G eenland empe a u es and he s a e o he No h-
e n Hemisphe e a mosphe ic ci cula ion we e subjec o
non-Gaussian noise, and i indeed pulses o his noise ig-
ge ed ansi ions be ween s adial and in e s adial egimes,
his would ha e impo an implica ions o ou concep ion
o s abili y o ce ain clima e ipping elemen s. The possi-
bili y ha clima e ipping elemen s a e nowadays likewise
subjec o non-Gaussian s ochas ic o cing wa an s mo e
a en ion.
6 Conclusion
In his wo k, we p esen ed a da a-d i en analysis o he
δ18
O and dus concen a ion eco ds om he NGRIP ice co e,
based on he K ame s–Moyal equa ion. This equa ion gen-
e alises he Fokke –Planck equa ion by allowing o a bi-
a ily complex unco ela ed d i ing noise
dξ
. In pa icula ,
such noise may esul in a discon inuous p ocess.
The es ima ion o he KM coe icien s yielded a monos-
able d i o he isola ed
δ18
O eco d and a bis able one o
he dus . The analysis o he esul ing agnos ic ime se ies
models does no allow o conclusions abou he dynamical
1 3
465 Page 8 o 14
Discon inuous s ochas ic o cing in G eenland ice co e da a
D
m(x)∼
1
m!
1
∆ ⟨(x +∆ −x )m|x =x⟩
∼1
m!
1
∆
1
N
N−1
∑
i=1
K(x−xi)(xi+1 −xi)m
.
(B1)
Simila ly o selec ing he numbe o bins in a his og am, o
a ke nel-densi y es ima ion, we selec bo h a ke nel and a
bandwid h (Nada aya 1964; Wa son 1964; Lamou oux and
Lehne z 2009). The ke nel is a unc ion K(x) o he es ima-
o

h(x)
, whe e h is he bandwid h a a poin x, ollowing

h(x)=
1
nh
n

i=1
K

x−xi
h

(B2)
o a collec ion
{xi}
o n andom a iables. The ke nel
K(x) is no malisable
´K(x)dx=1
and has a bandwid h
h, such ha
K(x)=K(x/h)/h
(Rydin Go jão e al. 2019;
Taba 2019; Da is and Bu e 2022). The bandwid h h is
equi alen o he selec ion o he numbe o bins, excep ha
binning in a his og am is always ‘placing numbe s in o non-
o e lapping boxes’. We use an Epanechniko ke nel
R
2=
N
∑
i=1 (
X i−aX∆T( i)−
{
bGI,i i∈GI
bGS,i i
∈
GS
)2
,
(A1)
wi h X ei he dus o
δ18
O and wi h espec o he pa ame e s
aX
,
bGI
, and
bGS
. Fo a gi en ime
i∈GS (GI)
indica es
ha
i
alls in o a s adial (in e s adial) pe iod. The esul -
ing
aX
is used o de end he o iginal da a wi h espec o
he empe a u e. The de ended da a a e subsequen ly no -
malised by sub ac ion o hei mean and di ision by hei
s anda d de ia ion.
Appendix B: Nada aya–Wa son es ima o o
he KM coe icien s and bandwid h selec ion
In o de o ca y ou he es ima ion in Eq. (6) we map each
da a poin in he co esponding s a e space o a ke nel den-
si y and hen ake a weigh ed a e age o e all da a poin s
Fig. 4 Remo al o a linea end in he NGRIP
δ18
O and dus ime
se ies (No h G eenland Ice Co e P ojec s membe s 2004) wi h espec
o a global a e age su ace empe a u e econs uc ion (Snyde 2016).
In panel (a) bo h o iginal
δ18
O (ligh b own) as well as de ended and
no malised (pu ple) ime se ies a e shown. Likewise o he dus eco d
in panel (b) (ligh b own and g een, espec i ely). The backg ound
empe a u e is gi en in anomalies wi h espec o he mean o e he
in es iga ed pe iod (blue). Panels (c) and (d) show sca e plo s o he
o iginal
δ18
O and dus da a wi h espec o empo a ily co esponding
empe a u e anomalies, espec i ely. Da a om in e s adials (s adials)
is shown in o ange (ligh blue). The black dashed lines co espond o
he i ing scheme ha uses a single slope bu wo di e en o se s o
sepa a ely i he s adial and in e s adial da a
1 3
Page 9 o 14 465