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\DOI{10.5802/crgeos.231}
\datereceived{2023-01-11}
\daterevised{2023-07-30}
\dateaccepted{2023-07-31}
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\dateposted{2023-09-20}
\begin{document}

\begin{noXML}

%\makeatletter
%\def\TITREspecial{\relax}
%\def\cdr@specialtitle@english{Geodynamics of continents and oceans - A tribute to Jean Aubouin}
%\def\cdr@specialtitle@french{G\'eodynamique des continents et des Oc\'eans - Hommage \`a Jean Aubouin}
%\makeatother

\title{Analogue model of rift linkage and inversion with application to
the Western Alps}

\author{\firstname{Nicolas} \lastname{Dall'Asta}\CDRorcid{0000-0002-1493-2672}\IsCorresp}
\address{Universit\'{e} de Pau et des Pays de l'Adour, E2S UPPA, CNRS, LFCR, France}
\email[N. Dall'Asta]{nicolas.dallasta@univ-pau.fr}


\author{\firstname{Yoann} \lastname{Den\`{e}le}\CDRorcid{0000-0003-1307-6330}}
\address{GET, Universit\'{e} de Toulouse, CNRS, UPS, IRD, CNES, 14 av. E. Belin, F-31400 Toulouse, France}
\email[Y. Den\`{e}le]{yoann.denele@get.omp.eu}


\author{\firstname{Vincent} \lastname{Regard}\CDRorcid{0000-0002-5250-6068}}
\addressSameAs{2}{GET, Universit\'{e} de Toulouse, CNRS, UPS, IRD, CNES, 14 av. E. Belin, F-31400 Toulouse, France}
\address{Department of Earth and Atmospheric Sciences \& School of Natural Resources, University of Nebraska-Lincoln, Lincoln, Nebraska, USA}
\email[V. Regard]{vincent.regard@get.omp.eu}


\author{\firstname{Anne} \lastname{Frayssignes}}
\addressSameAs{2}{GET, Universit\'{e} de Toulouse, CNRS, UPS, IRD, CNES, 14 av. E. Belin, F-31400 Toulouse, France}
\email[A. Frayssignes]{anne.frayssignes@univ-tlse3.fr}


\author{\firstname{Guilhem} \lastname{Hoareau}\CDRorcid{0000-0002-0603-5242}}
\addressSameAs{1}{Universit\'{e} de Pau et des Pays de l'Adour, E2S UPPA, CNRS, LFCR, France}
\email[G. Hoareau]{guilhem.hoareau@univ-pau.fr}


\author{\firstname{Sylvie} \lastname{Leroy}\CDRorcid{0000-0002-3188-8802}}
\address{Sorbonne Universit\'{e}, CNRS-INSU, ISTeP, Institut des Sciences de la Terre de Paris, France}
\email[S. Leroy]{sylvie.leroy@sorbonne-universite.fr}


\author{\firstname{Thibaut} \lastname{Pires}}
\addressSameAs{2}{GET, Universit\'{e} de Toulouse, CNRS, UPS, IRD, CNES, 14 av. E. Belin, F-31400 Toulouse, France}
\email[T. Pires]{thibautpires31@gmail.com}

\shortrunauthors

\keywords{\kwd{Rift linkage}
\kwd{Rift inversion}
\kwd{Inheritance}
\kwd{Analogue models}
\kwd{Strain field}
\kwd{Western Alps}}

\begin{abstract}
Along-strike segmentation of orogens raises questions because its
causes may predate orogeny in relation to structural inheritance. Here
we focus on rift/margin linkage domains and their inversion by using
analogue models with image analysis to extract the 3D strain field.
Extensional models document, depending on the strike-perpendicular
offset and the brittle-crust thickness, three types of rift linkage
modes: (1) oblique linkage with early T-fault, (2) strongly-oblique
linkage with R-fault network and, (3) transfer-linkage with late
Y-strike-slip fault. Analogue model of inverted rift basins is used to
analyse the misunderstood tectonic evolution of~transition zones in the
segmented~Western Alpine belt.
\end{abstract}

\maketitle

\twocolumngrid

\end{noXML}

\section{Introduction}

As the continental lithosphere is very heterogeneous, rifts are rarely
formed in straight lines but are more generally characterised by
associations of rift segments separated by rift linkage zones.
Numerical and analogue modelling highlighted a large variety of rift
linkage structures that can be subdivided into two main groups, the
accommodation zones (sometimes called relay ramps), in which
deformation is distributed and rift boundary faults do not connect, and
transfer zones in which a localised fault zone connects rifts segments
\citep[e.g.,][]{Acocellaetal2005,ZwaanSchreurs2017}. These two types
of interactions are characterised as ``soft linkage'' and ``hard
linkage'' \citep{WalshWatterson1991}, respectively.\unskip\break Several parameters
have been identified as controlling rift linkage type and the
development of specific structures. The large strike-perpendicular
offset between rift segments promotes transfer zones to be narrower
\citep{MauduitDauteuil1996,Dauteuiletal2002} or even prevent transfer
zone development leading to the formation of accommodation zones
\citep{VendevilleLeCalvez1995,Autinetal2010,Allkenetal2012,LePourhietetal2017,Neuharthetal2021}.
Accommodation zones development with overlapping geometry between rift
segments is favoured by orthogonal rifting \citep{ZwaanSchreurs2017}.
With increasing deformation \citep{Acocellaetal2005} earlier
accommodation zones can evolve into transfer zones through the creation
of a large variety of rift linkage structures. The nature of the
linkage structures is controlled in part by the overlap or underlap of
rift segments at the beginning of oblique fault development
[along-strike offset, \citealp{TentlerAcocella2010}]. Noticeably, the
rift linkage zone type depends also on the rift extension mode, whether
wide, with numerous normal faults on each side, or narrow, with a
restricted number of faults accommodating the deformation. The rift
extension mode is dictated by the rheology of the crust/lithosphere
\citep{Buck1991,Bucketal1999,Brun1999,HuismansBeaumont2007}. Wide rift
extension mode is favoured by a weak lithosphere that tends to prevent
transfer zones to develop \citep{Allkenetal2012,Zwaanetal2016}. In a
recent paper, based on high-resolution 3D numerical modelling,
\citet{Neuharthetal2021} show that the key factors controlling the
transition between hard and soft linkages are firstly the
strike-perpendicular offset ($x$-offset) and secondly the strength of the
continental crust \citep{Neuharthetal2021}. For a crustal configuration
typical of the continental interior with a ratio of 25~km upper and 
10~km lower crust, they define a typology of rift linkage in four
regimes as the strike-perpendicular offset increases, from oblique
linkage, transform linkage, microplate formation to rift jump. The
first two types are examples of hard linkage, the last type of soft
linkage. Microplate formation type, leading to the individualization of
microcontinents, is an example of an accommodation zone evolving to a
double transfer\unskip\break zone.

Mountain belts are particularly non-cylindrical, showing curved
structures with along-strike variations in the timing and kinematics of
major structures. These differences are related to variations in
convergence velocities and finite shortening but also reflect the
three-dimensional pre-structuration of the crust and lithosphere [i.e.
structural inheritance, \citealp{Mouthereauetal2013,Bellahsenetal2014}]. 
Large-scale orogenic curvatures in the Tethyan (or Alpine-Himalayan)
mountain belts, for example, such as the Alpine arc or the Himalayan
oroclines are known to reflect indentation processes of the Adria
microplate in Europe and Indian plate in Asia, respectively
\citep{Tapponnieretal1986,Rosenbaum2014}. On a smaller scale,
widespread evidences of along-strike segmentation seem to be partly
controlled by the inversion of the segmented passive margins of the
Neo-Tethys ocean and its oceanic branches such as the Alpine Tethys or
the Valaisan in the Alpine orogen
\citep[e.g.][]{LemoineTrumpy1987,Ribesetal2019}, as well as mature
rifting domains as the lower-Cretaceous Pyrenean rift 
\citep{Tugendetal2015,LescoutreManatschal2020}. In such orogenic belts,
the inversion of offset syn-rift segments leads to specific structures,
accentuating their non-cylindrical nature. Depending on the degree of
reactivation of normal faults and of the counterpart propagation of
thrust, specific structures, for instance footwall shortcut, break back
thrust and thrust stacking of extensional imbricates, develop
\citep[e.g.][]{Boninietal2012}. The 3D strain field of such systems
needs to be investigated in a way to better restore and thus understand
orogenic systems. Structural inheritance related to offset syn-rift
segments inversion is indeed recognized to be a first-order parameter
controlling Alpine-type orogenic growth
\citep[e.g.,][]{Mouthereauetal2012,Bellahsenetal2014}, but is also one
of the less known effects, often neglected for the sake of\break
simplification.

In this paper we focus on the formation of offset rift basins but also
more broadly of offset margin domains that form early during the
rifting (proximal to inner necking zones). We are also concerned with
inversion of these two types of systems during mature collision
processes by accretion of poorly thinned continental crust subsequently
to both early inversion of hyper-thinned rift or margin domains during
immature collision (i.e. soft collision) and subduction processes which
consumed the oceanic lithosphere. The purpose of this paper is to
analyse the 3D strain field during extension and then inversion of
offset rift segments with brittle analogue \mbox{models.} The
Pelvoux--Mont-Blanc rift segments
\citep{Lemoineetal1989,Ribesetal2019,DallAstaetal2022} in the Alpine
orogen, will be compared to the models' extracted 3D strain fields
during extension and inversion. By assuming that the 3D deformation
field of rifts and poorly thinned margin domains and their inversion
during mature collision phases is controlled by frictional processes
involving the upper crust decoupled from underlying ductile processes,
this cross analysis allows to integrate the first-order parameters that
control the pre- to syn-orogenic evolution of the upper-brittle
rheological level of orogens.


\section{Model setup}

We performed analogical experiments to determine the influence of
kinematics on the geometry and localization of structures in rift
linkage zones and the structural response when inverted during orogenic
shortening. The design of the reference model in extension is inspired
by the paleo-geodynamic reconstructions of the Western Alpine realm
during lower-Jurassic early stages of rifting 
\citep{Lemoineetal1989,Pfiffner2014,Ribesetal2019}. In these
reconstructions, extension is defined to be orthogonal to rift segments
and rift linkage zones at high-angle to the direction of rift segments,
with $x$-offset less than 100~km (case of the Mont-Blanc/Belledonne
transition zone) and greater than 100~km (case of the
Pelvoux/Belledonne transition zone). Previous analogue modelling
studies have shown that a high linkage angle (angle between
segments and rift linkage) is favoured in the case of small
along-strike offset \citep{TentlerAcocella2010}, defined here as
$y$-offset (Figure~\ref{fig1}). That explains why we have chosen systems
without underlapping or overlapping geometries between rift segments
(e.g. $y$-offset $=$ 0). The transient brittle--ductile transition is
considered at a depth of 15~km with respect to previous studies of the
Alpine Tethyan rifted margins \citep{Mohnetal2012}. The ratio
length/thickness of the rift linkage domains is thus smaller or higher,
respectively, than a value of~6.7.

\begin{figure*}
\includegraphics{fig01}
\caption{\label{fig1}(A) Map view of the experimental set-up, only the dark
grey area is moving. (B) Top photograph of the NDA02 experiment before
the extension (note the passive grid of 5~cm wide tiles at the top of
the model). (C) 3D view of the experimental set-up showing the
colourful patterns of the sand pile. (D)~Qualitative strength profiles
in the model (left) and nature (right), BDT: Brittle--ductile
transition.}
\end{figure*}

To model the rift linkage zones during extension and inversion, we
consider rift segments orthogonal to the divergence and then
convergence directions and rift linkage zones are initially set up
parallel to the displacements. The experiments consider the response of
a 3~cm thick brittle layer to the deformation imposed at its base. The
scaling factor with the Alps is thus 3~cm in the model representing 
15~km in nature.

In the experiments, the length of the rift linkage zones is variable
(10 and 20~cm, corresponding to length/thickness ratios of 3.3 and 6.7,
respectively). These experiments are conducted on a large experimental
table above which a metal plate is pre-cut into segments and rift
linkage (Figure~\ref{fig1}); the displacement was generated and
controlled by a computer-driven stepper motor that pulled (or pushed
during inversion) the plate at a constant rate.

We do not present all models here for simplicity, but rather illustrate
the evolution of rift linkage domains from a representative model. To
model the brittle crust, whose behaviour is of Mohr-Coulomb type
\citep{Byerlee1978}, we use dry sand layers (quartz sand MI 0.1/0.35
from Sibelco; average grain size 0.26~mm (friction angle
${\sim}32\text{\textdegree}$, sand density 
${\sim}1.55~\mathrm{g}/\mathrm{cm}^{3}$) for a total thickness of 
3~cm. The inside layers have been coloured by adding colourant, which
does not change the mechanical properties (Figure~\ref{fig1}D). A
passive grid of 5~cm wide tiles has been drawn on top of the sand pile
(Figure~\ref{fig1}B). The basal metal plate is shaped to create two
rift linkage zones of length 10 and 20~cm, separating extension zones
lying perpendicularly (Figure~\ref{fig1}). The extension zones are
20~cm long (central basin), and ${>}$20~cm on both sides
(outer basins), which prevents boundary effects. The model was extended
by 4~cm before being inverted and shortened by 5~cm.

Model evolution was documented by top photos (Figure~\ref{fig1}B),
shot every 0.5~cm of deformation (0.25~cm in some cases). The
characteristics of the top photos are unchanged throughout the
experiment. The photographs were all rectified in the same way as the
initial photo, which serves as a reference. For this one, the
rectification was based on the grid drawn previously. We then
documented the strain thanks to the tools PIVlab 
\citep{ThielickeStamhuis2014} and Strainmap \citep{Broerseetal2021}.

\section{Results of analogue modelling}

We describe in the following the reference experiment (label NDA03),
but with additions from experiment NDA02, for cross-section
interpretations (NDA02 is similar to NDA03, without inversion).


\subsection{Map view evolution of rift linkage zones during extension}

The first strain increments lead to the opening of three basins
bordered by two master normal faults in the rift segments. These
conjugated normal faults are orthogonal to the extension direction. The
structural evolution of the rift linkage zones occurred in three steps
whose characteristics depend on the $x$-offset. In the initial step,
deformation is diffuse. In the later steps, we observe two types of
kinematic regimes: Type 1, connection through a strongly-oblique
linkage zone; Type 2, connection through a transfer linkage zone. These
two types are described below, following observations reported in
Figure~\ref{fig2}.

\begin{figure*}
\includegraphics{fig02}
\caption{\label{fig2}Interpreted pictures with stretching lineation directions
extracted from strain map analysis (see Supplementary Material),
principal strain axis stretch  $\lambda\max$ ($\lambda  = 1$ represents
no length change), rotation  $\omega$  (negative for clockwise et
positive for anticlockwise) for the reference model NDA03. The dotted
lines represent inactive faults.}
\end{figure*}


\subsubsection{Strongly-oblique linkage}

After the initial step of diffuse deformation, the connection between
the rift segments is visible on~the surface of the model from 0.5~cm of
extension. It takes place first in a localised manner on a Riedel-type
oblique master fault (R-fracture \citep{Riedel1929}, with an angle of
${\sim}30\text{\textdegree}$ with the direction of extension, 
Figure~\ref{fig3}), which creates a bridge between the
inner parts of the rift segments. R-type control of the incremental
bridge results in the formation of a linkage zone that is strongly
oblique to the rift segments with a linkage angle of around
60\textdegree. We subsequently notice a progressive distribution of the
deformation. This takes place first by development at the tip of the
rift segments of multiple convex faults with both transtensional and
strike-slip segments. The connection between the R-type
master oblique fault and the multiple convex faults that spreads
progressively inside, associated with clockwise/anticlockwise rotations
that affect the outside part of rift segment tips, tends to enlarge the
growing linkage zone and increase the linkage angle; those parameters
reach 2~cm and 70\textdegree\ for 4~cm of extension. Strain is
controlled by simple shear in the growing linkage, as shown by a highly
oblique stretching direction (at an angle of
30\textdegree--60\textdegree) to the extension direction. Subsidence is
initially strongly localised on the R-type fracture and the inside part
of rift segment tips, then becomes more diffuse from 1.75~cm of
extension with the propagation of depression areas from the curved tips
of rift segments into the interior of the bridge, to cover the entire
strongly-oblique rift linkage domain after 4~cm of\unskip\break extension.

\begin{figure*}
\includegraphics{fig03}
\caption{\label{fig3}Rotation $\omega$ (negative for clockwise et positive for
anticlockwise) and displacement field in the two rift linkage zones
that develop in the NDA3 experiment.}
\end{figure*}

\subsubsection{Transfer linkage}

The incipient bridge (0 to 0.5~cm of extension) between rifts segments
is recorded by diffuse simple shear in the centre of the rift linkage
zone (Figures~\ref{fig2} and~\ref{fig3}). Multiple convex faults then
develop (from 0.5~cm of extension) at the outside tips of the rift
segment. From 1~cm of extension two master R-type oblique faults form
from the inside tip of rifts segments and get connected from 1.25~cm of
extension in the centre of the linkage domain by an incipient Y-type
fault (i.e. parallel to the extension). From 1.5~cm of extension the
deformation tends to localise on the master Y-fault that grows from the
centre to the borders of the linkage domain; the Riedels and convex
faults are left inactive. The final strike-slip structure is not
exactly perpendicular to the rift segments: its strike differs by a few
degrees (${\sim}$12\textdegree). As in the strongly-oblique
linkage domain, strain is controlled by simple shear and marked by a
highly oblique stretching direction. The subsidence history is marked
first by minor curved depression areas in the inside tips of the
segment and a large topographically high centre domain that develops
until 1.5~cm of extension in a transpressive restraining bend localised
between the two R-type faults. From 1.75~cm of extension, a localised
depression alignment develops along the Y-fault trend which intersects
the topographically high domain which persists on the edges of the
fault up to 4~cm of extension.


\subsection{Map-view evolution during inversion}

Shortening only leads to the partial closure of the 
basins (Figure~\ref{fig4}), as new
thrusts develop outside the basins before basin closure
(${\sim}$3~cm of shortening). One consequence is that the
depressions corresponding both to inherited rift segments and rift
linkage zones are not removed and last up to the end of the experiment
(5~cm of shortening).

The structural evolution during inversion occurs in two steps whose
characteristics vary between initial rift segments and rift linkage
zones. The first step is marked by passive folding related to
distributed shortening in the core segments (Figure~\ref{fig4}).
The strongly-oblique
linkage inversion occurs by a combination of neoformation of R-type
dextral faults and inversion of the inherited convex faults
(Figure~\ref{fig4}).
Interestingly, a rotation of up to 15\textdegree\ of the strike-slip
fault is visible during shortening (clockwise for inherited
strongly-oblique linkage, anticlockwise for inherited transfer
linkage). At the same time, the strike-slip zone tends to become less
localised. The transfer linkage zone is affected by sinistral
strike-slip motion localised in the inherited dextral Y-type fault,
associated with transpressive deformation in the inherited convex
multiple faults at the tip of inherited rift segments (Figure~\ref{fig4}).
This deformation
regime leads to the formation of a peculiar horsetail fault geometry
incompatible with the sinistral strike-slip motion that occurred during
\mbox{inversion.} The second step of shortening takes place \mbox{after}\break
ca. 3~cm of
inversion and is associated with the \mbox{neoformation} of in-sequence thrust
systems on both sides of the core segments and neoformed thrust-related
lateral ramps centred on the inherited rift linkage domains
(Figure~\ref{fig4}). These
lateral ramps gradually impose a widening of the inherited linkage zone
by outward development, particularly visible at the end of the
experiment for the strongly-oblique\unskip\break linkage.

\begin{figure*}
\includegraphics{fig04}
\caption{\label{fig4}Interpreted pictures with stretching lineation directions
extracted from strain map analysis (see Supplementary Material),
principal strain axis stretch $\lambda\max$ ($\lambda = 1$ represents
no length change), rotation $\omega$ (negative for clockwise et
positive for anticlockwise) for the reference model NDA03 during
inversion.}
\end{figure*}


\subsection{Cross sections}
\subsubsection{Core segments}

The extension is accommodated in core segments by a main normal fault
(50\textdegree) that develops on the mobile metal plate and antithetic
faults in its hanging-wall, in the fixed part of the sand pile; both
accommodate only dip-slip motion. The antithetic faults develop in
sequence as the basal metal plate moves; it leads to an asymmetric rift
basin. While the main normal fault is active throughout the whole
extension, each antithetic fault accommodates only part of the
extension (Figure~\ref{fig5}A).

\begin{figure*}
\includegraphics{fig05}
\caption{\label{fig5}Line drawing of the cross sections after a 4~cm extension
(panels A, C, E, experiment NDA02) and after a 5~cm inversion (panels
B, D, F, experiment NDA03) with the addition of the photograph of the
model for panel (B). The cross sections were cut at similar locations and
the difference between the two cross sections of the same pair is
assumed to represent the evolution during inversion. Green: faults
active during extension; red: faults active during shortening; light
green: reactivated faults. Numbers represent the time sequence of fault
activation. Pair A and B: cross sections in core segments; pair C and
D: cross sections in the middle of the strongly-oblique linkage;~ pair
E and F: cross sections in the middle of the transfer linkage. The
original photographs of the cross sections are available in
Supplementary Figure~5.}
\end{figure*}

During inversion, the master normal fault is first reactivated (1 in
Figure~\ref{fig5}B). The inner part of the core segment is squeezed in
the middle against the major fault as shown by buttress folding. Then,
when shortening increases, synthetic reverse faults form parallel to
the main normal fault (2, 3, Figure~\ref{fig5}B). This constitutes the
short limb of~ an asymmetric inverted rift basin. The long limb is
marked by shortcuts preserving part of the inherited normal faults
($1'$~and~$1''$) and subsequent outward propagation of thrusts ($2'$,~4,
Figure~\ref{fig5}B).


\subsubsection{Rift linkage zones}

Cross sections show steep faults branching at depth with small vertical
displacements: they correspond to flower structures, whether positive
or\unskip\break negative (case shown Figures~\ref{fig5}C and~E). The
movement accommodated is mostly lateral, invisible in cross section and
interpreted from map-view photographs.

The inversion in the inherited strongly-oblique linkage zone led to the
formation of an imbricate structure composed of 3 thrust units
(Figure~\ref{fig5}D). These thrust units (1, 2 and 3) are smaller
than in the core zone (Figure~\ref{fig5}D). An apparently almost
vertical fault accommodates most of the dextral strike-slip motion in
the unit 1. Interestingly, parts of the steep faults inherited from the
extension stage are preserved and transported passively on the nappes
(units 1 and 2).

Within the inherited transfer linkage, it is interesting to note that
faults developed during extension are clear: the outer faults are
Riedel while steep faults near the centre are Y-planes
(Figure~\ref{fig5}E). The cross sections suggest that inversion mostly
led to the reactivation of the Y-plane while Riedels show only minor
reactivation (Figure~\ref{fig5}F). A lateral ramp is observed
connecting the two main neoformed thrusts that develop in the fixed
sand pile part of the core segment and accommodating most of the
deformation in the transfer zones after 3~cm of inversion. These
different sections show that preserved rift-related structures can be
observed locally in the inherited rift linkage domains.

\section{Discussion of the model}

To better understand the strain field in fossil rift and margin systems
now included in mountain belts we focus this part of the discussion on
the results of the extensional models. The results of the model in
inversion will be discussed by comparison with the Alpine case in the
Western Alps. The main results of the models for the transfer zones
(e.g., strain directions, subsidence evolution) are then applied to the
Western Alpine case to try to highlight the influence of rift-related
pre-Alpine transfer zones on the segmentation of the Alps.

In extension, for $x$-offset lower than 200~km (equivalent to 40~cm in
our model) we reproduce the transition highlighted in numerical
modelling, from oblique linkage to transfer linkage with a transfer
zone almost orthogonal to rift segments \citep{Neuharthetal2021}. Our
results also clearly show a decrease in the width of the linkage domain
with increasing $x$-offset \citep{MauduitDauteuil1996}. Although the
transfer linkage zone is similar to that obtained by 
\citet{Neuharthetal2021}, the oblique linkage zone is significantly
different in our reference model, with a larger initial linkage angle
(60\textdegree\ vs. 25\textdegree) and a subsequent contrasting
structural evolution. The linkage angle was recognized as being
dependent on the first-order on the $y$-offset by 
\citet{TentlerAcocella2010} but this conclusion is challenged by the
results obtained by \citet{Neuharthetal2021} which show very similar
rift linkage domains for the $y$-offset between 100 and 400~km.
Furthermore, it must be considered in natural examples that the
$y$-offset is only a function of the propagation time of the rift
segments, which is not the case for the $x$-offset.
\citet{LePourhietetal2017} and  \citet{Neuharthetal2021} suggest a
first-order control on the linkage angle by the crustal strength. The
results of numerical models by \citet{Allkenetal2012} with a brittle
upper crust overlying a ductile lower crust show that hard linkage is
favoured against soft linkage in the case of strong decoupling. For the
case of weak coupling with a brittle crust decoupled from underlying
deformation (closely similar to our models), these results suggest a
first-order control on the linkage angle of the ratio (called here
$x_{h}$ \textit{ratio}) between the $x$-offset and the thickness $h$ of
the brittle\unskip\break crust.

To better understand the control on linkage angle in the case of low
$x$-offset (${<}$200~km) we have tested the impact of various $x_{h}$
\textit{ratios} in other experiments, named NDA1 and NDA4 which are
presented in Supplementary Material and synthesised in
Figure~\ref{fig6}. Our results show a clear control of the $x_{h}$
\textit{ratio} on the linkage angle. For low $x_{h}$ \textit{ratio}
(${<}$2.5) we reproduced the characteristics of the rift linkage
(linkage angle of ca. 35\textdegree) obtained by
\citet{Neuharthetal2021} for low $x$-offset: the linkage mode is oblique,
controlled by the early development of T Riedel-type planes leading to
a rapid connection between normal faults of the rift segments. For
intermediate $x_{h}$ \textit{ratio} (2.5 to 4.5), strongly-oblique
linkage development is controlled by the early formation of R
Riedel-type planes. Finally, for a higher $x_{h}$ \textit{ratio} 
(${>}$4.5), transfer-linkage development is controlled by the later formation
of a Y Riedel-type fault. These 3 linkage modes, which are
characteristic of low $x$-offset (${<}$200~km) in decoupled crust, can be
explained by a competition between simple shear accommodating
deformation in the rift linkage zone and local stress variations around
the main normal fault. For instance, the coseismic slip on normal
faults is associated with a decrease of stress on the footwall and
hanging-wall \citep{NuchterEllis2011,ThompsonParsons2016}, and a stress
increase on the fault extremities, which can possibly drive fault
propagation \citep{KingDeves2015}. It appears that for a low $x_{h}$
\textit{ratio}, the local stress effects dominate and drive the
propagation of the rift segments, favouring the formation of T-planes
linking the two rift segments (Figure~\ref{fig6}). For a higher $x_{h}$
\textit{ratio}, the rift linkage is dominated by simple-shear tectonics
marked by the development of R and Y planes. Riedel shear fracture
planes form and connect the two rift segments. The interaction between
the Riedels and the main normal fault leads on the extremities to the
rotation of the Riedel and the formation of convex faults. If the
$x$-offset is too important, the Riedel faults can no longer connect the
segments and therefore, the linkage will be achieved through the Y
plane\unskip\break (Figure~\ref{fig6}).

\begin{figure*}
\includegraphics{fig06}
\caption{\label{fig6}(A) Results of the NDA1 experiments ($h = 8.5$~cm, $x$-offset $=$
10~cm, $x_{h}$ \textit{ratio} $=$ 1.2) after 1~cm and 4~cm of
extension, principal strain axis stretch $\lambda\max$ is represented
for the interval 0.5--1~cm of extension, (B) Results of the NDA4
experiment after 4~cm of extension ($h = 3$~cm, $x$-offset $=$ 1, 2, 3,
5, and 15~cm, $x_{h}$ \textit{ratio} $=$ 0.3, 0.7, 1, 1.7, and 5), (C)
Synthesis of the NDA experiment with zooms representing the evolution
of the $\omega$ parameter to illustrate first-order structures
developing in rift linkage domains and structural schemes illustrating
the evolution of the various rift linkage zones defined in this study
(see text for explanations).}
\end{figure*}


Our results, in accordance with high-resolution numerical models
\citep{Allkenetal2012,Neuharthetal2021}, show a first-order control
of the $x$-offset and the $x_{h}$ \textit{ratio} on the mode of linkage.
The three types of linkage modes defined here show contrasting records
in terms of tectonic evolution and subsidence history. These results
based on analogue modelling provide clues to better understand the
structural record and the stratigraphic architecture of the basins of
the highly-segmented fossil margins. Oblique linkages are marked by a
rather rapid subsidence history (i.e., after a small amount of
extension) with a depocenter that extends both from the centre of the
linkage zone on normal faults (T-factures) and the tips of the
segments. This type of linkage is also characterised by highly oblique
faults that limit sub-basin domains and a stretching directed at 20\textdegree\ to
30\textdegree\ to the extension direction. The strongly-oblique linkage
and transfer zones are marked by a very oblique stretching to the
extensional direction (60\textdegree\ to 90\textdegree) attesting to
strain control by simple shearing. Strongly-oblique linkage is marked
by a network of two sets of faults, some with purely strike-slip
kinematics and slightly oblique to the direction of extension and
others with convex shapes and transtensional kinematics.
Strongly-oblique linkage is marked by a depocentre that propagates from
the segment tips towards the core which becomes a topographic low after
about 2~cm of extension. In the transfer zones, the subsidence history
is radically different, the linkage zone undergoes strong uplift up to
4~cm of extension, and only a small area intersected by the Y fault is
in subsidence and only after 4~cm of extension. The structural record
shows the interaction of 3 types of faults, purely strike-slip faults
oblique to the direction of extension or parallel to the direction of
extension and faults with a transtensional component with convex
shapes.

\section{Comparison of analogue models with the Western Alps
(Pelvoux--Mont-Blanc\newline segment)}

This section discusses how our models provide a better understanding of
the Pelvoux--Belledonne--Mont-Blanc segment in the Western Alps. Two
possible rift linkage domains are described: the Pelvoux--Argentera
transfer zone to the south and the Belledonne/Mont-Blanc transition
zone to the north. The existence of these linkage domains is difficult
to assess due to the strong Alpine deformation obliterating most of the
rift-related structures. The~now inverted rift linkage domain between
the Belledonne and Mont-Blanc External Crystalline Massifs is
characterised by syn-rift stretching lineations highly-oblique to the
direction of extension whose tectonic meaning remains unclear
\citep{DallAstaetal2022}. We bring in the following new insights from
analogue modelling.


\begin{figure*}
\vspace*{2pt}
\includegraphics{fig07}
\vspace*{5pt}
\caption{\label{fig7}(A) Paleogeographic maps of Western Europe showing the
location of the Liguro-Piemont and Valais oceanic basins during the
Late Jurassic  \citep[based on][]{Decarlisetal2017}. (B) Proposal of a
paleogeographic reconstruction of the Western Alps during the Late
Jurassic showing the inferred transform systems in oceanic basins
aligned with continental rift linkage domains  
\citep[based on][]{LemoineTrumpy1987,Lemoineetal1989,Ribesetal2019,DallAstaetal2022}.
Abbreviations for the External Crystalline Massifs: Ar: Argentera, 
Pv: Pelvoux, Bd: Belledonne, M-B: Mont-Blanc. (C) Cross-section of
the Valais and Liguro-Piemont basins at the end of Jurassic
\citep[modified from][]{Mohnetal2012}.}
\vspace*{4pt}
\end{figure*}


\subsection{Geological setting}


The Jurassic rifting in the Western Alpine realm is associated with the
opening of two oceanic basins (Liguro-Piemont and Valais basin see
Figure~\ref{fig7}) separated by a continental ribbon (Brian\c{c}onnais
block). The area of interest corresponds to the
Pelvoux--Belledonne--Mont-Blanc segment (ca. 170~km long,
Figure~\ref{fig7}B) of the inverted European margin bounded to the
south by the inferred Pelvoux Argentera transfer/transform fault
\citep{Lemoineetal1989} and to the north by the inferred Aar-Mont-Blanc
transfer/transform fault. These faults correspond to syn-rift transfer
zones and syn-spreading transform faults. Inside the Pelvoux/Mont-Blanc
segment a lower-Jurassic rift linkage zone is proposed between the
Mont-Blanc and Belledonne massifs \citep{Pfiffner2014,Ribesetal2019}.
Rifting is recognized to be associated with NW--SE extension direction
\citep{Lemoineetal1989,LeBretonetal2021}. While the distal part of the
European and Adria margins have been widely studied, the transition
between the proximal domain ($25 \pm 5$~km-thick continental crust) and
the hyperextended domain (${<}$10~km continental crust) corresponding
to the necking domain remain poorly constrained in the Western Alps
(Figure~\ref{fig7}C). It was proposed that the External Crystalline
Massifs (ECM) and surrounding basins correspond to the former necking
domain of the European margin \citep{Mohnetal2014}.

The Jurassic margin was inverted during the Late Cretaceous 
\citep{Fordetal2006}. Inversion started with a subduction initiated in
the Liguro-Piemont domain before a second subduction zone was 
created~in~the Valais during the Eocene \citep{RosenbaumLister2005}.
Propagation of the deformation toward the continent led to the burial
of the necking domain (e.g. ECM) under the accretion prism during the
Late Eocene and Oligocene  
\citep{SchmidKissling2000,SimonLabricetal2009,Cardelloetal2019},\unskip\break
followed by thick-skin deformation of the ECM during Oligocene and
Miocene times \citep{HerweghPfiffner2005, Bellahsenetal2012}. The
shortening directions in the ECM are E--W to NW--SE 
\citep{Gillcrist1988,Fordetal2006}, and orthogonal to the margin
\citep{Gillcrist1988}. Finally, late-stage NE--SW dextral strike-slip
deformation in the ECM in relation to Late-Miocene Apulia rotation led
to a clockwise rotation~of the massifs
\citep{Gourlay1984,HubbardMancktelow1992}.


\vspace*{-1pt}

\subsection{Comparison with models}

\vspace*{-1pt}
\subsubsection{Extension}


In the External Crystalline Belledonne massif, Jurassic high-angle
NE-striking normal faults have been recognized 
\citep{Lemoineetal1981,BarfetyGidon1984,Chevalieretal2003}. Although
minor N--S extension is recorded in the massif, most extension is
accommodated by two Late Liassic (Sinemurian) NE--SW faults, the Ornon
and Mizoen faults \citep{Gillcrist1988}. The Ornon fault is the best
preserved and most studied of these faults (Figure~\ref{fig8}C and D).
It shows a NW--SE-directed syn-rift stretching direction
\citep{Gillcrist1988,Chevalieretal2003}, i.e. parallel to the direction
of extension. Further north, Jurassic rift-related structures in the
Aiguilles-Rouges/Mont-Blanc massif are more complex. The external
Mont-Blanc corresponds to the termination of the Morcles basin where
the sedimentary cover is pinched out \citep{Epard1986} while the
internal Mont-Blanc likely corresponds to a Jurassic detachment fault
[Figure~\ref{fig8}F, \citealp{Ribesetal2020, DallAstaetal2022}]. 
In the northern
part of the detachment fault, stretching lineations are NW--SE-oriented
(Figure~\ref{fig8}B), parallel to the main extension direction and
orthogonal to the normal faults as observed in the Western Alps.

\begin{figure*}
{\vspace*{1pt}}
\includegraphics{fig08}
{\vspace*{2pt}}
\caption{\label{fig8}(A) Result of the analogue modelling for an extension of 1~cm.
(B) Results of the analogue modelling for an extension of 4~cm. (C)
Paleogeographic reconstruction of the Pelvoux--Mont-Blanc segment during
the Late Jurassic  \citep[based
on][]{Lemoineetal1989,Gourlay1984,Ribesetal2019}. The stretching
directions are from \citet{Gillcrist1988} and \citet{DallAstaetal2022}.
(D) Cross section of the Ornon fault \citep[modified
from][]{Gillcrist1988}. (E) Cross section of the Beaufortain area
\citep[modified from][]{Epard1990}. (F) Cross section of the Mont-Blanc
massif.}
{\vspace*{1pt}}
\end{figure*}


Unlike these areas which are textbook cases (e.g., Ornon fault),
unexplained strain fields can be observed. At the transition between
the Mont-Blanc and Belledonne massifs for example, the southern part of
the detachment fault shows a N--S stretching direction
[Figure~\ref{fig8}C; Col du Bonhomme; \citealp{DallAstaetal2022}],
oblique at an average of 45\textdegree\ to the NW--SE direction
\citep{Lemoineetal1989}. In addition, the detachment fault is crosscut
by NW--SE Jurassic high-angle normal faults showing a strike slip
component. In this Mont-Blanc/Belledonne \mbox{transition,} the Beaufortain
area is marked by specific rift-related structures, which were proposed
by \citet{Butler1985} and \citet{Epard1990} to be extensional
imbricates forming small half-graben basins accommodating minor
extension oblique to the main normal faults (Figure~\ref{fig8}C and E).
This fault architecture differs intriguingly from the central segments,
both in terms of fault direction and offset. In this case, the analogue
models we performed allow us to understand all the observations without
the need to appeal to an exotic tectonic phase as an alternative
hypothesis. Indeed, our analogue models strongly suggest that the
oblique orientation of the stretching directions is one of the main
characteristics of the oblique and transfer linkage modes.
Interestingly, in the model, strongly-oblique linkage is characterised
by the formation of R-type and convex faults connecting the two main
segments. These observations indicate that the area localised between
the northern Belledonne and southern Mont-Blanc might correspond to a
strongly oblique linkage zone with a connection that bridges the Ornon
fault and the Mont-Blanc detachment fault (Figure~\ref{fig8}C).
Noteworthy, a delay in the subsidence is observed between the Ornon
fault and the Belledonne/Mont-Blanc transition based on the detailed
stratigraphic description from the Ornon cover and Beaufortain area. On
the Ornon fault (Belledonne segment), significant subsidence is
observed during Hettangian and Sinemurian times [ca. 110~m/Ma,
\citealp{Chevalieretal2003}]. On the contrary, in the transition
between the Belledonne and the Mont-Blanc, the subsidence was much
lower until the Pliensbachian (ca. 13~m/Ma between Hettangian and
Sinemurian). We observed that R-linkage is similarly associated with a
delay in subsidence (Figures~\ref{fig6} and~\ref{fig8}A and B) compared
to the main segments. This further suggests the importance of the
oblique linkage understanding from analogue models to explain the
Belledonne--Mont-Blanc transition.\looseness=-1

The southern termination of the Belledonne/{\ubreak}Mont-Blanc segment
corresponds to the Pelvoux--Argentera transform fault (Figure~\ref{fig8}C).
Although the kinematics of this domain is not well constrained, it was
proposed by 
\citet{Gillcrist1988}, based on the inversion of Jurassic structure,
that the termination of the transform was connected to the Ornon fault
by a convex fault network forming the tip splay of the transform fault.
The analogue models support this hypothesis: they display similar convex
faults, on the tip of the transform segments making the connection with
the main segments. Moreover, similarly to what is observed in models,
the Pelvoux area underwent uplift during rifting \citep{Rouxetal1988}
which could be associated with the development of a transpressive
restraining bend along the future transform fault.

In such Alpine Tethys examples, the initial localisation of the main
rift segments ($x$ and $y$-offset) might be related to the pre-rift
structure of the crust (e.g., inherited crustal weakness). It is
noteworthy that the Belledonne--Mont-Blanc (Beaufortain) linkage zone
coincides with a change of basement lithology, from micaschist in the
Belledonne massif to gneiss/granite in the Mont-Blanc massif, implying
a different rheological behaviour of the basement crust.


\subsubsection{Inversion}

Some of the deformations observed in the inverted transfer zones of the
model can be tentatively recognized in the Western Alps and could
explain some of the early compressional features in the Beaufortain.

\begin{figure*}
\includegraphics{fig09}
\caption{\label{fig9}(A) Result of the analogue model showing the inversion of the
core segment and linkage zones after 5~cm of shortening. (B) Simplified
structural map of the Western Alps, focusing on the Beaufortain area
between the Belledonne and Mont-Blanc massif. A-R $=$ Aiguilles-Rouges,
M-B $=$ Mont-Blanc. The Maures massif located further south west of the
Pelvoux is not represented. (C) Cross section of the core segment of
the model. (D) Cross section of the Belledonne massif 
\citep[from][]{Bellahsenetal2014}. (E)~Cross section of the inverted growing
linkage in the model. (F) Cross section of the Beaufortain area
[modified from \citealp{Eltchaninoff1980}, and \citealp{Butler1985}]. (G) Evolution of
the External Crystalline Massifs (ECM) from the end of the rifting to
the convergence \citep[based on][]{Gourlay1984,Bellahsenetal2014}. The
collision is associated with the rotation of the ECM, the closure of
the Morcles Basin and the inversion of the transfer zone in the
Beaufortain and Pelvoux massifs.}
\end{figure*}

In detail, the reactivation/inversion of the core segments is well
illustrated by the Bourg d'Oisans half-graben of Jurassic age
(Figure~\ref{fig9}B and D). The temperatures reached in the Belledonne
massif during Alpine shortening favoured the development of large shear
zones at ca. 30--25~Ma \citep{Bellangeretal2015}. These led to the
closure of the half-graben and the extrusion of the syn-rift cover. In
this case, the Jurassic Ornon fault acted as a buttress as highlighted
by the strong vertical NE--SW foliation visible in the sediments close
to the fault plane
\citep[][Figure~\ref{fig9}D]{Bellahsenetal2014}.
The deformation in the cover was mostly accommodated by flattening
against the Jurassic fault. Similar deformation can be observed in the
Morcles basin between the Aiguilles-Rouges and Mont-Blanc massifs
(Figure~\ref{fig8}C), where it resulted in the extrusion of the
syn-rift cover and the formation of the NE-striking Morcles nappe
\citep{Ramsay1981}. The comparison with analogue models suggests that
pre-existing normal faults were only marginally inverted, and rather
had a role of buttress (Figure~\ref{fig9}C and D). This indicates a
strong fault misorientation with respect to the inversion stress field.
It also led to failure of the footwall forming short-cuts, and to the
formation of break-back thrust in the hanging-wall
\citep[][Figure~\ref{fig9}]{BoyerElliott1982}. It should be noted that
the models seem to reflect well the natural case, in spite of the
plastic deformation which prevailed in nature after the ECM (External
Crystalline Massifs) were heated to a temperature higher than
300~\textdegree C following their burial under the internal nappes
\citep{Bellangeretal2015}. 

The inversion of the linkage between the main segments is more complex.
As proposed in the previous section, the Beaufortain might correspond
to a strongly-oblique linkage domain connecting the Ornon fault and the
Mont-Blanc detachment fault. Early deformation in the Beaufortain is
marked by the presence of numerous thrusts oblique to the NE--SW thrust
in the Mont-Blanc and Belledonne massifs, and showing N--S stretching
directions associated with dextral strike-slip movement along the
faults (Figure~\ref{fig9}B). It corresponds to an early deformation
associated with S1 foliation \citep{Epard1990}. It was noticed by
\citet{Epard1990} that these faults do not cross-cut late Alpine
folds, indicating that they are inherited and/or were active during
early Alpine deformation, as also confirmed by ages as old as 14 Ma for
N--S dextral shear in the Beaufortain \citep{Eglietal2017}. These faults
constitute a nappe imbrication fan, contrasting with the Alpine
structures organised into a crustal antiformal stack
(Figure~\ref{fig9}F). Some of these small nappes also correspond to an
inverted Jurassic half-graben [e.g., Sallestet syncline; \citealp{Epard1990}].
In the area, late NE--SW strike-slip deformation occurred between the
Mont-Blanc and Aiguilles-Rouges 
\citep[since ca. 9~Ma][]{Bergemannetal2019}, driving significant
clockwise rotation       
\citep[Figure~\ref{fig9}G;][]{Gourlay1984,Collombet2001,Delacouetal2008}. The
models we carried out allow us to overcome this late history. They
teach us that the nappe imbrication, oblique with respect to the main
direction of extension then shortening, is a completely natural process
if we consider that this zone is a rift linkage zone. This oblique
deformation would therefore be contemporary to, and thus compatible
with extension followed by shortening on the main segments considered
here (Belledonne and Mont-Blanc).

Finally, we give a last example to convince that the rift linkages can
express very oblique strain fields with respect to the directions of
the surrounding structures: the Pelvoux--Argentera massifs. Indeed, the
inversion of the transform in the Pelvoux led to the inversion of the
splay in the Pelvoux massif associated with the formation of sinistral
strike-slip faults \citep{Gillcrist1988}. In addition, NW--SE lateral ramps
(Figure~\ref{fig9}B) connect the NE--SW thrust system of the Belledonne massif to
thrusts systems of the Maures massif to the southwest.

Paired analogue models/structural analyses observations show that rift
inheritance, especially in the rift linkage zones, can lead to complex
kinematics during inversion. Some elements observed in the model such
as the highly-oblique stretching in the linkage zone as well as the
oblique faulting and subsidence can be observed in the Western Alps.
This is particularly the case for the Beaufortain, highlighting the
possible presence of a Jurassic transfer zone between the Mont-Blanc
and Belledonne massifs. These observations might explain early Alpine
deformations in the Beaufortain area, which still remain poorly
understood. These simple analogue models might give new insight to
better understand offset rift margins in the Alpine Tethys and their
consequence on the subsequent collision.

\section{Conclusion}

In this study we show based on extensional analogue modelling that rift
linkage between rift segments in the case of small $x$-offset 
(${<}$200~km) in a brittle upper-crust mechanically decoupled for
underlying structural levels, can occur in three modes. The
oblique-linkage mode is characterised by the rapid connection between
rift segments on a transtensional fault zone, and early subsidence
history. The strongly-oblique linkage mode is marked by a progressive
bridge formation by connections of strike-slip and convex
transtensional faults, and a later history of subsidence. The
transfer-linkage mode is driven by the late formation of an orthogonal
strike-slip fault and characterised by the formation of a large
uplifted transpressional domain. We find that the $x$-offset and the
ratio between the $x$-offset and the brittle crust thickness (called here
$x_{h}$ \textit{ratio}) is the primary factor determining the
connection mode. We also show that inversion of rift linkage domains
leads to specific structural characteristics in inverted rift linkage
domain combining the preservation of some inherited strike-slip faults,
the formation of short-wavelength nappe stack decapitating early
transtensional faults and neoformation of preferentially strike-slip\break
faults.

When compared to the Alpine case, the analogue models succeed in
illustrating how the main normal faults inherited from the rifting
phase behave during inversion as buttresses squeezing post-rift
sediments. But more significantly, the models address the problem of
oblique deformation directions observed towards the terminations of the
main rift segments. The models show us that the evidence of deformation
oblique with respect to the main direction of extension then
shortening, is a completely natural process if we consider that these
zones are rift linkage zones. Such an oblique deformation is
unquestionably compatible with the extension and inversion of the main
segments, whose structures are inherited from the rifting phase. These
results might give new insight to better understand offset rift margins
and their consequence on subsequent collision not only in the Alpine
Tethys but in other orogens\break worldwide.


\section*{Conflicts of interest}

Authors have no conflict of interest to declare.


\section*{Acknowledgements}

This work benefited from the financial support of E2S-UPPA to the first
author. We benefit from the analogue modelling lab set up by F. Odonne
\citep[e.g.][]{Dufrechouetal2011}, whom we warmly thank. Benjamin
Guillaume and Gianreto Manatschal are thanked for initiating us to
PIVLAB and Strainmap analyses and discussions about the Alpine case,
respectively. We would like to thank also the reviewer Franck Chanier
and the editor Laurent Jolivet for the comments which helped to improve
this manuscript.

\back{}

\section*{Supplementary data}

Supporting information for this article is available on the journal's
website under \printDOI\ or from the author.  

\CDRsupplementaryTwotypes{supplementary-material}{\cdrattach{crgeos-231-suppl.pdf}}

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