The model layers comprise a granular “upper crust”, ductile “lower crust”, and ductile “lithospheric mantle”. The ductile materials exhibit spatially continuous deformation at the scale of observation. They behave viscously under our range of experimental strain rates, simulating deformation in the viscous layers of the lithosphere (i.e. the lower crust and lithospheric mantle). The brittle–ductile layers are isostatically supported by a liquid that is analogous to the natural asthenosphere (Fig. 2a).

The yield strength profiles of the models resemble natural lithospheric strength profiles. The model strength profiles include a relatively strong upper crust as well as lower crust and lithospheric mantle layers of varying relative strengths (Fig. 2e; Table 1). As the experiments were designed to help us better understand Proterozoic craton-wide rifting in the North Australian Craton (Allen et al., 2015), we implemented a rheological layering that allowed extension to be relatively uniform across the entire model area and create a distributed system of basins (i.e. “wide rifting” sensu Buck, 1991; also see Brun, 1999, and Buck et al., 1999). Hence the model lithosphere is analogous to a natural thick lithosphere (with a thick crust) shortly after orogenesis or with a higher-than-normal heat flow (Buck et al., 1999). Given the challenge of reconstructing the lithosphere configuration and rifting conditions of the North Australian Craton in the Proterozoic, we used the Basin and Range Province – a well-known example of a wide rift (e.g. Hamilton, 1987; Parsons, 2006) – as a proxy for estimating crustal thicknesses (Gueydan et al., 2008) and the rate of extension for our models. Hence, the thicknesses of the crustal layers in Models R1 and R2 scale to 10 and 40 km for the upper and lower crust respectively. After running Models R1 and R2, we found that it would be more representative of the North Australian Craton (Betts et al., 2002; Kennett et al., 2011) to have upper and lower crust layers with the same thickness, which we then implemented in Models R3, R4, and R5 (Table 1).

The models were extended at a velocity that scales to 1–2 cm yr-1 in nature (Table 2), which is within the range of estimated rates of extension for the Basin and Range Province (Bennett et al., 1998; Snow and Wernicke, 2000; Hammond and Thatcher, 2004; Tetreault and Buiter, 2018). After ∼ 8 cm of extension (∼ 20 % bulk extension), the models were shortened in the reverse direction at the same rate until they reached their initial pre-extension length (Fig. 2), simulating orthogonal rifting and then shortening. Before the start of extension, dark coffee powder was sifted onto the light-coloured model surface, which created high-contrast speckle patterns for deformation monitoring (Sect. 2.3). More coffee powder was sifted after the end of extension to ensure that there were sufficient dark particles inside the newly formed rift basins. As it took time to add coffee powder to the model surface, ensure that there were no problems with recording during extension, and initiate the shortening part of the experiment, 7–24 min elapsed between the end of extension and the start of shortening. This pause scales to a maximum of ∼ 1.7 Myr in nature (not considered significant compared to the entire duration of extension and shortening; see Sect. 2.2 for details on scaling factors). Model R1 is an exception, as no shortening was applied after extension.

The model lithosphere layers were created using granular and ductile materials similar to those used by Molnar et al. (2017) and Samsu et al. (2021). The brittle upper crust, the behaviour of which can be described using the Mohr–Coulomb law (Byerlee, 1978), was modelled using a granular mixture comprising 89 % dry quartz sand (Rocla 90 Fine Foundry Sand, Hanson Australia) and 11 % hollow ceramic Envirospheres^®. The sand is fine-to-medium grained, with 78 % of the grain sizes falling between 150 and 425 µm. The Envirospheres^® were added to the sand to ensure that the density of the model upper crust is lower than that of the lower crust (Table 2) and still scales appropriately to the natural upper crust (cf., 2670 kg m-3; Artemjev and Kaban, 1994). Using a Hubbert-type shear box, we measured an internal friction angle ϕ ∼ 49∘ and maximum extrapolated cohesion C ∼ 120 Pa for the granular mixture. These values are high compared to a similar but slightly coarser-grained mixture used by Molnar et al. (2017; ϕ < 38∘ and cohesion C ∼ 9 Pa; 75 % of grains between 435 and 500 µm), the latter of which seemed more consistent with the ≤ 60∘ dip of the normal fault surfaces from the extensional phase of the experiments. Therefore, we adopted the internal friction angle and cohesion of the granular mixture used by Molnar et al. (2017) for the calculation of differential stress in the brittle layer of the strength profiles.

The ductile lower crust and lithospheric mantle layers were modelled using mixtures that mostly consist of polydimethylsiloxane (PDMS). For the lithospheric mantle, black Colorific^® plasticine was added to the PDMS in order to increase its effective viscosity (Boutelier et al., 2008). On its own, PDMS is a Newtonian fluid (Weijermars, 1986), meaning that its viscosity is strain rate independent. However, the PDMS-based mixtures used to model the lithospheric mantle in our experiments are slightly non-Newtonian, as they exhibit strain rate-softening behaviour (stress exponent n ∼ 1.4; Table 2).

Iron filings were also added to the lower crust and lithospheric mantle material to increase their densities. For Models R4 and R5, the addition of fine-grained iron filings (0.42–0.82 mm grain size, manufactured for ChemSupply Australia) did not significantly affect the flow behaviour of the PDMS-based mixture during our experiments (LC2 and LM2 in Table 2). However, in Models R1, R2, and R3, the use of ultrafine-grained iron filings (manufactured for Mad About Science), which has a powder-like consistency, had the unintended effect of doubling the viscosity of the PDMS-based mixture (compare LC1 and LC2 in Table 2). We opted to use fine-grained iron filings for Models R4 and R5 to mitigate this viscosity increase. Hence the yield strength profile for Model R3 contains a “strong” lower crust and high LC:LM strength ratio, while Models R4 and R5 contain a lower crust that is significantly weaker than the lithospheric mantle (low LC:LM strength ratio; Fig. 2e, Table 1); the latter is more consistent with theoretical strength profiles for a wide rift setting (Brun, 1999).

The scaling parameters (Ramberg, 1967) used in our experiments and the properties of the model layers are presented in Table 2. These parameters were chosen so that model deformation is consistent with natural processes but occurs over a timescale that is convenient for laboratory experiments. The length scaling factor L* = Lm/Lp= 4 × 10-7 meant that 0.4 cm in the model scales to 10 km in nature, whereby the subscripts “m” and “p” denote the model and natural prototype respectively. Therefore, the model surface area of 44 × 40 cm corresponds to 1100 × 1000 km in nature, and the model thicknesses of 2.8–3.2 cm represent lithospheric thicknesses of 70–80 km. The density scaling factor ρ* = ρm/ρp was set to 0.46 based on the density ratio between the model asthenosphere material (Queen Glucose Syrup; Schellart, 2011) and estimated asthenosphere densities between 3100 and 3400 kg m-3, consistent with previous lithospheric-scale analogue experiments (e.g. Molnar et al., 2017; Santimano and Pysklywec, 2020; Samsu et al., 2021) and reference asthenospheric densities used in geophysical models (e.g. 3250 kg m-3 in Lamb et al., 2020). Similarly, the viscosity scaling factor η* = ηm/ηp was determined using the ratio between the effective viscosity of the model asthenosphere (520 Pa s) and that of the natural asthenosphere (1.9 × 1019 Pa s for Models R1 and R2; 1.9 × 1020 Pa s for Models R3, R4, and R5). As the experiments were conducted under normal gravitational acceleration, the scaling factor for acceleration due to gravity g* = gm/gp= 1, resulting in a stress scaling factor σ* = ρ* × g* × L* between 1.7 × 10-7 and 1.9 × 10-7. The above scaling factors were used to calculate the time scaling factor t* = η*/(ρ*×g*×L*) and velocity scaling factor v*=L*/t*.

For Models R1 and R2, we started out with a natural asthenosphere density ρp= 3100 kg m-3 and viscosity ηp= 1.9 × 1019 (following Molnar et al., 2017) and an extension velocity that scaled to 2 mm yr-1, resulting in an extension duration of 14 h. For Model R3, the objective was to explore the behaviour of the ductile layers when we extended the model by the same amount but at a faster rate. Therefore, the prototype viscosity was increased by 1 order of magnitude (to p= 1.9 × 1020) to achieve an appropriate time scaling factor. This change in the time scaling factor enabled us to apply an extension rate that still scaled to 2 mm yr-1 in nature within a shorter (experimental) extension duration, i.e. around 3 h. However, additional changes to the ductile materials were still necessary, as the strength contrast between the lower crust and lithospheric mantle (LM1 in Table 2) in Model R3 was too low to simulate the natural lithosphere with a strong lithospheric mantle and relatively weak lower crust (Table 1, Fig. 2c). Therefore, for Models R4 and R5, we created an improved lithospheric mantle mixture (LM2 in Table 2) with the desired viscosity ηm= 2.7 × 105 Pa s (approximately 10 times greater than the model lower crust), resulting in a low LC:LM strength ratio. As this mixture had a density ρm= 1384 kg m-3, the density scaling factor was changed to 0.42 (using ρp= 3400 kg m-3 for the asthenosphere), otherwise the prototype lithospheric mantle and asthenosphere densities would have both equalled 3100 kg m-3. This last change did not significantly impact the other scaling factors. The layer densities in Models R4 and R5 are the most consistent with those used in geophysical models on the density structure of the lithosphere, where the densities of the upper crust, lower crust, and lithospheric mantle are 2700, 2940, and 3350 kg m-3 respectively (Kaban et al., 2014).

The difference in the experimental strain rates between Models R1 and R2 and Models R3, R4, and R5 is a consequence of the difference in the viscosity scaling factor v* and therefore the time scaling factor t*. The time scaling factor for Models R1 and R2 (1.5 × 10-10) is 1 order of magnitude higher than for R3 (1.5 × 10-11) and R4 and R5 (1.6 × 10-11). The experimental strain rate for Models R1 and R2 (∼ 5.4 × 10-5 s-1) is 1 order of magnitude lower than for Models R3 (∼ 3.1 × 10-4 s-1) and R4 and R5 (∼ 2.8 × 10-4 s-1). These strain rates were estimated by dividing the rate of extension (i.e. the velocity of the moving wall during extension) by the initial thickness of the model lithosphere (Benes and Scott, 1996). As the strength of the ductile layers increases with the applied strain rate (Ranalli, 1995; Brun, 1999), the lithospheric mantle in Models R1 and R2 is weak compared to the lithospheric mantle in R3, R4, and R5 (Fig. 2d). For R1 and R2, the strength profile shows that the lower crust and lithospheric mantle are both weak compared to the upper crust, due to the slow strain rate applied to these experiments.

The scaling parameters used in Model R3 and in Models R4 and R5 are relatively similar, with the main difference being the effective viscosity of the lower crust in R3 being 2 times greater than in R4 and R5 (Table 2). As a result, in Model R3 the lower crust is almost as strong as the lithospheric mantle (Fig. 2). In R4 and R5, the lower crust is much weaker than the lithospheric mantle. The relative strength of the lower crust with respect to the overlying and underlying layers affects the mechanical coupling between the upper crust and lithospheric mantle and therefore the strain distribution in the upper crust, as discussed further in Sect. 3.1.

Digital image correlation (DIC) was applied to sequential images of the model surface in order to monitor deformation in the cover layer during the experiment. This technique allowed us to observe the strain and topographic evolution of the models. Strain maps and orthorectified photographs of the model surface were used to track the formation of rift basins and inversion structures at different stages of the experiments.

The image acquisition and DIC workflow is similar to that outlined in Molnar et al. (2017) and Samsu et al. (2021). The DIC system comprises two cameras at oblique angles to the model surface (Fig. 2a). Images were recorded at 5 min intervals over 14 h for experiments R1 and R2 and at 2 min intervals over approximately 3 h for experiments R3, R4, and R5 for each extension or shortening phase. Surface strain and topography were computed using the StrainMaster module of the commercial image correlation software DaVis (version 10.1.2, LaVision). The software uses stereo cross correlation to compute the incremental displacement field from which the strain tensor components derived. In our experiments, high-contrast speckle patterns created by coffee powder sifted on the model surface were used for image correlation.

For the strain maps, the displacement vector fields obtained from DaVis were used to derive incremental and cumulative axial strain (eyy and Eyy respectively) in MATLAB. eyy and Eyy are measures for normal strain parallel to the extension and shortening direction. Eyy was computed from the displacement gradient tensor. The displacement gradient tensor H comprises the components ΔDiΔxj, where Di are the displacement components in the x and y direction (i.e. Du and Dv) and xi refer to the x and y axis of the coordinate system (with the y axis being parallel to the extension and shortening axis). Using the Lagrangian finite strain tensor E (Allmendinger et al., 2011): 1E=ExxExyEyxEyy. The normal strain along the extension and shortening axis can be calculated with 2Eyy=12ΔDuΔy+ΔDvΔy+ΔDuΔyΔDuΔy+ΔDvΔyΔDvΔy.

The incremental vertical displacement (dw), cumulative vertical displacement (Dw), and the height (i.e. topography) of the model surface were also calculated in DaVis.

The incremental and cumulative displacement field data generated by DaVis were imported into MATLAB for post-processing for visualisation purposes. Post-processing of data included detection and replacement of spurious displacement vectors and interpolation of missing data. To this end, we used the discrete cosine transform and partial least squares (DCT-PLS) algorithm (Garcia, 2011). Topographic data were corrected by fitting a plane through the initially flat but possibly tilted model surface. Finally, the resulting linear correction parameters were applied to all subsequent digital elevation models. The MATLAB scripts used for post-processing and visualisation are available at https://github.com/TimothySchmid/PIV_postprocessing_2.0 (last access: 20 February 2023).

In all of our experiments, the imposed bulk extension of the model resulted in an extension-orthogonal, E–W trending horst and graben system. Here we define “graben” as a topographic depression bounded by parallel normal faults (Reid et al., 1913; Peacock et al., 2000) which accommodate the bulk extension imposed on the models (Figs. 3, 4, 6). We also use the terms “graben” and “basin” interchangeably. The wide distribution of basins is analogous to natural examples of wide rifting, such as in the Proterozoic North Australian Craton (Allen et al., 2015; Betts et al., 2008), Basin and Range Province (Wernicke et al., 1988), Aegean Sea (Doutsos and Kokkalas, 2001), and East China Rift System (Tian et al., 1992).

In the experiments, rift evolution occurred in two main phases: (1) rift basin formation associated with normal fault nucleation, growth, and linkage and (2) basin deepening and widening. Basins became fully established when the normal basin-bounding faults reached their final length. Progressive extension allowed the throw along the faults to increase, resulting in deepening of the grabens. Downward fault propagation was limited by the thickness of the upper crust, after which strain was accommodated by widening of the grabens.

The timing of the above processes with respect to amount of bulk extension varied with different model setups (see initial strength profiles and boundary conditions in Fig. 2e and Table 1). The rift system evolved most quickly in Models R4 and R5, followed by R3 and then R1 and R2. For example, the model topography shows that by 10 % bulk extension, graben-bounding faults in Models R4 and R5 appear to have already reached their full lengths, while the faults in Models R1, R2, and R3 are still in the nucleation and growth stage (cf. Figs. 3, 4).

The models also exhibit different degrees of strain distribution, in terms of the spacing of faults and grabens, at the end of the extensional phase (at 19 %–20 % bulk extension; Figs. 3, 4). Grabens in Models R1 and R2 are evenly spaced across the model area. Their spacing is greater compared to Models R3, R4, and R5. They are highly segmented, bounded by normal faults with short and irregular (non-linear) fault traces. These grabens are also relatively shallow, due to the thin upper crust layer (Table 1).

Fault traces in Models R3, R4, and R5 are more clearly defined than in Models R1 and R2, due to the lower quality of particle image velocimetry (PIV) data for Models R1 and R2 (possibly caused by the sparser distribution of tracking particles on the model surface and non-optimal lighting conditions). Grabens in Model R3 are narrower and more segmented (i.e. less laterally continuous along the graben axes) than those in Models R4 and R5. Models R3, R4, and R5 exhibit clusters of grabens that make up so-called “high-strain zones”, which are separated from each other by “low-strain zones”, where the distance along the y axis between two adjacent basins is greater than the widest basin in the model (Fig. 5). Hence, basins are not distributed uniformly across the model area as they are in Models R1 and R2. There is no strain in the northernmost and southernmost ∼ 5 cm of the model, as these segments were attached to the confining plexiglass walls; these areas are not included in the subsequent analyses. In the north of the model, there is a narrow low-strain zone adjacent to a wide high-strain zone, the latter of which makes up approximately the southern two-thirds of the model area. In Models R3 and R4, another low-strain zone is present within the wide high-strain zone (Fig. 5).

Deformation of the extended lithosphere is accommodated by brittle faulting in the upper crust and viscous flow of the lower crust and lithospheric mantle. In our experiments, an initial period of distributed extension was followed by the localisation of deformation onto rift-related normal faults, which controlled the formation of rift basins (Figs. 3, 4, and 6). The wide distribution of basins is consistent with previous extensional experiments of brittle–ductile models in which a rift seed was not implemented (e.g. Benes and Davy, 1996; Gartrell, 1997; Corti, 2005), which would have otherwise localised rifting from the onset of extension (i.e. “narrow rifting” in Buck, 1991). While all our models demonstrate wide rifting, different degrees of mechanical coupling between the model layers appear to have influenced the details of rift evolution (i.e. timing of basin formation) and the overall distribution of faults and basins.

In a wide rift mode, regular spacing between basins reflects the characteristic wavelengths of periodic instabilities during extension. These instabilities require a strength or viscosity contrast between two or more layers, and they form uniformly spaced domains of greater thinning, known as boudinage or pinch-and-swell structures (Ramberg, 1955; Smith, 1977). The formation of rift basins is controlled by two different wavelengths of periodic instabilities, which occur at a smaller scale in the brittle upper crust (i.e. crustal boudinage) and at the whole-lithosphere scale (i.e. lithospheric boudinage; e.g. Benes and Davy, 1996). It is possible that the characteristic wavelengths of deformation localisation in our experiments is a product of the superposition of crustal and lithospheric boudinage, given their brittle–ductile multi-layer setup. In Models R3 and R4, zones of localised lithospheric necking may correspond with high-strain zones, while areas that underwent minimal stretching may correspond with low-strain zones (Figs. 4 and 5). Here, the distances between the centres of high-strain zones may represent the characteristic wavelength of lithospheric-scale boudinage. In Model R4, the spacing between high-strain zones is approximately 110 mm. The initial thickness of the ductile layers is 20 mm, giving a dominant wavelength / thickness ratio of 5.5, which is higher than the analytically predicted ratio of ∼ 4 for lithospheric necking (Smith, 1977; Fletcher and Hallet, 1983). This disagreement may partly be due to the low stress exponent of our ductile layers (n≤ 1.4) and the influence of the overlying upper crustal sand layer. In Models R3, R4, and R5, basin spacing is of the order of the initial crustal thickness (18 mm). Benes and Davy (1996) observed a similar relationship between basin spacing and crustal thickness in their wide rift analogue experiments, from which they interpreted that the characteristic wavelength of crustal-scale periodic instabilities is of the order of the crustal thickness.

The coupling between the layers in analogue experiments is controlled by the relative strengths (i.e. effective viscosities) of the ductile layers, which is in turn influenced by the rate of extension (Zwaan et al., 2021; Brun, 1999). The rate of extension in Models R1 and R2 was much slower than in the other models (Table 1), so that the lithospheric mantle was relatively weak and underwent uniform thinning during extension. The thick and weak lower crust in R1 and R2 had sufficient time to flow during extension. As a result, both ductile layers thinned over a wide region, so that strain was distributed evenly in the overlying upper crust, resulting in evenly spaced basins from north to south (cf. Benes and Davy, 1996). The wide spacing between the basins could be attributed to the large ratio of upper-to-lower crustal strength (Fig. 2e), which predicts deformation localisation in a few structures (Brun, 1999) and large spacing between basins, with each basin-bounding fault taking up a relatively large amount of strain (Wijns et al., 2005; Corti, 2005).

The faster rate of extension in Models R3, R4, and R5 resulted in a relatively strong lithospheric mantle compared to R1 and R2 (Fig. 2e; cf. Brun, 1999; Nestola et al., 2015). This strong lithospheric mantle is overlain by a strong ductile lower crust in R3 and weak ductile lower crust in R4 and R5. Therefore, coupling between the lithospheric mantle and brittle upper crust is stronger in Model R3 than in Models R4 and R5. In previous experiments by Gartrell (1997), the brittle upper crust was underlain by a strong, high-viscosity ductile layer (i.e. a so-called stress guide) and weak ductile lower crust. In their experiments, necking instabilities developed in the strong ductile layer and localised deformation into rift basins in the directly overlying upper crust. Similarly, our Model R3 consists of a strong ductile lower crust that directly underlies the upper crust and is almost as strong as the lithospheric mantle. Hence, the tight spacing between basins in Model R3 may correspond to short-wavelength localisation instabilities in the strong lower crust and lithospheric mantle. In Models R4 and R5, the weak lower crust acted as a decoupling layer between the strong upper crust and strong lithospheric mantle. While this decoupling by an intervening weak layer does not appear to significantly influence the spacing between basins, it may have contributed to the formation of more laterally continuous basins of Models R4 and R5, as opposed to the short and segmented basins of Model R3 (cf. Benes and Davy, 1996). In any case, all of the experiments presented here demonstrate wide rifting, as predicted by Brun (1999) for a dominantly ductile lithosphere (brittle–ductile thickness ratio between 0.1 and 0.4; Table 1).

The evolution of axial strain in our models show that pre-existing rift basins exert a strong control on deformation related to far-field shortening. As normal basin-bounding faults formed in the upper crust during extension, they became zones of dilation within an otherwise undisturbed granular layer (Bellahsen and Daniel, 2005; Sassi et al., 1993; Mandl et al., 1977); these became pre-existing zones of weakness that were reactivated in a reverse sense in the early stages of shortening (Figs. 6 and 7). The reverse reactivation of weakened normal faults during basin inversion has also been observed in previous analogue (e.g. Marques and Nogueira, 2008) and numerical experiments (e.g. Buiter et al., 2009). Steeply dipping normal faults with a ∼ 60∘ dip are normally considered non-optimally oriented for dip-slip reactivation during subsequent shortening (e.g. Koopman et al., 1987; Brun and Nalpas, 1996; Eisenstadt and Sims, 2005). However, Bonini et al. (2012) suggested that reverse dip-slip reactivation is indeed possible if a pre-existing steeply dipping normal fault is rotated to shallower dips during shortening and/or the principal stress axes are rotated. Stress rotation may be caused by shearing at the base of the brittle crust, so that the maximum principal stress axis deviates from the horizontal and its angle with the pre-existing fault plane is reduced. It is possible that in our models, reactivation in the brittle crust is facilitated by the shallowing of the normal fault dips due to the upwelling of the underlying viscous material. In addition, shearing at the interface between the brittle and ductile layers and the presence of lateral heterogeneities after the extensional phase may have contributed to localised rotation of the principal stress axes.

As shortening progressed, basins became narrower (Fig. 7). These basins correspond with areas of previously thinned lithosphere (Fig. 1), which would have been weaker than the rift shoulders. Continued shortening resulted in inversion of the basins, which we interpret as having been driven by anticlinal folding of the ductile layers, based on observations of uplifted lower crust underneath the inverted basins (following the removal of upper crustal material at the end of the experiments). This interpretation is comparable with observations from analogue experiments of continental collision (Sokoutis and Willingshofer, 2011) and intraplate compression (Dombrádi et al., 2010), where strain is accommodated and topography is controlled by folding of pre-existing weak zones. We also interpret the anticlinal folding to have been facilitated by upward buoyancy forces where the upper crust (and therefore the lithosphere) was thinnest, in order to achieve isostatic equilibrium (Fig. 11). This upwelling of viscous material underneath thinned crust or lithosphere has been observed in previous analogue models of rifting (Allemand and Brun, 1991; Brun and Beslier, 1996; Nestola et al., 2015; Beniest et al., 2018; Zwaan and Schreurs, 2023).

In their thermomechanical experiments of basin inversion, Sandiford et al. (2006) found that inversion is localised in the centre of the basin due to higher-than-average thermal gradients beneath the basin centre. In our analogue models, we observed that inversion is also greatest in the basin centre, based on maps of cumulative vertical displacement (Fig. 7). Even though we cannot directly observe the influence of heat flow in our isothermal experiments, we speculate that at the end of extension, the basin centres in our models lie above the thinned parts of the model lithosphere. In nature, lithospheric necking corresponds to a reduction in rift strength as hotter asthenospheric mantle material replaces colder and stronger lithospheric mantle (Chenin et al., 2018). In our models, lithospheric necking (Sect. 4.1) allows the weaker model asthenosphere to replace the stronger lithosphere due to isostasy. Combining the results of Sandiford et al. (2006) and our experiments, we suggest that basin inversion in nature occurs where the lithosphere is weakest, and this zone of weakness is created by the thinning of the strongest layers of the lithosphere and the upwelling of hot asthenospheric mantle material.

The interpretation of thinned lithosphere facilitating inversion may be inadequate for explaining why only some basins were inverted in Models R3, R4, and R5 while others remained basins (Figs. 8, 9, and 10). In contrast, all of the pre-existing basins in Model R2 were inverted. Here we discuss factors that may have influenced (1) whether all or only some rift basins are inverted during subsequent shortening (i.e. comparing Model R2 with Models R3, R4, and R5) and (2) the periodicity of selective basin inversion (i.e. comparing Model R3 with Models R4 and R5).

Model R2 was extended and shortened at a rate that was 5 times slower than Models R3, R4, and R5 (Table 1). Therefore, the ductile lower crust and lithospheric mantle layers of Model R2 were much weaker than the brittle upper crust. This meant that the ductile layers would have thinned and thickened uniformly during extension and subsequent shortening respectively. With no strain localisation in the ductile layers, rift-related faulting and basin formation in the granular, brittle upper crust during extension would have been controlled by the localisation of brittle deformation (Figs. 10b and 11a). During shortening of Model R2, upper crustal deformation may have been driven by the reverse reactivation of basin-bounding faults. Here there may have also been the significant influence of upwelling of ductile material underneath thinned crust, driving basin inversion. Despite the thinness (∼ 4 mm) of the brittle upper crust in Model R2, the basins subsided enough to allow a locally thinned crust to form, creating a crustal weakness that is exploited for inversion.

The periodicity of basin inversion is only apparent when the models were extended and shortened at a sufficiently fast rate (i.e. Models R3, R4, and R5), which we interpret as having promoted localised viscous deformation as opposed to uniform thinning and thickening (i.e. Model R2). The inverted basins in Models R3, R4, and R5 may be underlain by uplifted (presumably folded) ductile lower crust. We assume that these anticlinal folds which were spaced evenly apart (Fig. 10) represent periodic instabilities with a characteristic wavelength λV (i.e. the distance between two anticlines; Figure 11c). Hence λV also corresponds to the distance between two inverted basins. Based on our models, we conclude that for a system of distributed basins, where the distance between basins (λB) is shorter than λV, only some basins will be inverted (Fig. 11d). Models of biharmonic folding (i.e. decoupled folding of the upper crust and lithospheric mantle, with two different wavelengths; Cloetingh et al., 1999) support the idea that topography can be controlled by the superposition of strain localisation at different levels of the lithosphere, with different characteristic wavelengths.

The wavelength and amplitude of folds during layer-parallel shortening is controlled by the thickness and rheology of the folded layers (e.g. Schmalholz and Mancktelow, 2016). While it is outside the scope of this work to analyse in detail how λV is influenced by the model setup (e.g. layer thicknesses, viscosity ratios, bulk strain rates), we introduce here some simple calculations to assess whether folding of the ductile lithosphere layers is plausible. By treating the combined ductile lower crust and lithospheric mantle as a single ductile layer resting on a homogeneous viscous medium, we estimate an Argand number (Ar) of ∼ 3 for Models R3, R4, and R5, using (Schmalholz et al., 2002) 3Ar=ΔρgH2μeffε˙B, where Δρ is the density difference between the liquid asthenosphere and air, g is the gravitational acceleration, H is the combined thickness of the lower crust and lithospheric mantle, μeff is the average effective viscosity of the lower crust and lithospheric mantle, and ε˙B is the experimental strain rate. An Ar of this magnitude suggests that any folding that would occur in our models would be controlled by gravity. In the gravity-controlled mode of folding, the maximal amplification rate (i.e. growth rate) of folds αgrav can be calculated from (Schmalholz et al., 2002) 4αgrav=6nArε˙B, with n being the power law exponent of the ductile layer. In Models R3, R4, and R5, the calculated growth rate varies between 0.0006 and 0.0007, which is approximately 2 times greater than the experimental strain rate. Hence it is theoretically possible for gravity-controlled folding to have occurred in the experiments (Schmalholz et al., 2002; Fig. 11d). In contrast, folding was theoretically insignificant in Model R2, which is consistent with a calculated Ar ∼ 17 and a maximum amplification rate of 2.2 × 10-5 (approximately 2 times less than the experimental strain rate). For the latter case, shortening was most likely accommodated by homogeneous layer thickening (Fig. 11b) or inverse boudinage (Zuber, 1987).

The selective inversion of rift basins in our models has not been observed in previous crustal- and lithospheric-scale analogue experiments. There are few other analogue experiments in which extension followed by shortening is applied to a brittle–ductile model during the same experimental run. Examples of such experiments include the work of Gartrell et al. (2005) and Cerca et al. (2010), where extension is followed by shortening in a direction that is oblique to the extension direction (by 10 and 15∘ respectively). However, these experiments differed from ours in that extension was localised by initial zones of weakness. As a result, the rift basins were not distributed across the entire model area, and all of the extensional basins localised subsequent shortening and associated inversion structures.

Our experiments show the importance of conducting lithospheric-scale analogue experiments – with a brittle–ductile multi-layer model underlain by a liquid asthenosphere for isostatic support – to investigate rheological controls on basin inversion. Future investigations on selective basin inversion would need to take into account sedimentation, as the density of basin sediments may suppress folding and uplift of the ductile layers during shortening. The lack of model cross sections in this investigation, which are challenging to make for a three-layer model lithosphere resting on glucose, hinders direct observations of important processes that are invoked, such as lithospheric boudinage and the upwelling of ductile and liquid material. Imaging of the ductile layers during experimental runs (e.g. using a CT scanner; Colletta et al., 1991; Schreurs et al., 2003; Zwaan et al., 2018, 2020; Zwaan and Schreurs, 2023) would allow us to better track viscous deformation, which, as we have shown, plays a significant role in promoting basin inversion.