​​​​​​​For this study, three models (I, II, III) were prepared with a similar base plate setup (Fig. 1) at room temperature (22 ∘C and humidity of 50 %–60 %). Models were initially 8.3 cm thick, 30 cm wide, and 40 cm long. A basal metal plate was attached to the moveable backstop that created a velocity discontinuity in the middle of the model (20 cm from the backstop) beneath the layers of a sand–magnetite mixture (Fig. 1). The sand–magnetite mixture was used to simulate the brittle behaviour of sedimentary rocks in the upper crust and consist of loose sand and magnetite (< 0.1 vol %), both with similar subangular shape and grain size (0.124–0.356 mm). The average bulk susceptibility of the sand–magnetite mixture is 1.9 × 10-3 [SI]. This indicates that the artificial high content of ferromagnetic multi-domain magnetite in the models (compared to natural examples) governs the bulk signal of the AMS and the influence of the diamagnetic sand can effectively be neglected. Single layers of the sand–magnetite mixture were carefully sieved from a height ranging between 30–50 cm above the model. The layers, which were sieved to varying thicknesses (0.9–1.5 cm) due to manual sieving, were separated by thin layers of coloured sand acting as passive markers. The rationale behind sieving the sand–magnetite granular mixture into the sandbox and accepting irregularities in layer thickness throughout the model was to avoid scraping, which has proven to create an artificial initial magnetic fabric (e.g. Schöfisch et al., 2022). The uppermost layer of each model consisted of sand only and was scraped after sieving to create an even model surface with the same model height for all three models. No samples were taken from this uppermost layer for the AMS analysis. On the surface of the models, coffee powder was sieved, and point markers (coloured sand) were set for monitoring surface deformation. Monitoring surface deformation is used for comparing the models and their development but is less crucial for the outcome of this study. The sidewalls of the sandbox model were transparent glass walls that allowed the monitoring of the model evolution during extension and inversion from the sides.

All three models (I, II, III), which were identical in setup, were extended up to 1 cm in total. However, only models II and III were inverted. During extension, the model runs were paused after 0.5 cm of extension to fill the developed basin. After the extension phase, models II and III were shortened by different amounts of bulk shortening to simulate basin inversion at different stages. Extension and shortening of the models were initiated by a constant moving backstop with a velocity of 3 cm h-1, to which a velocity discontinuity was attached. To describe orientation clearly, the backstop of the model was labelled as the geographic model, North. Model I simulated extension only. Model II was shortened by 1.5 cm after extension and stopped when the first kinks developed (i.e. when sand layers showed millimetres of offset monitored through the transparent sidewalls of the sandbox), before any thrust forms with larger displacement. Model III was shortened by 4 cm after the initial extension. This 4 cm amount of shortening represents the stage when the backthrust of Model III showed a similar amount of displacement as the pre-existing normal fault. Moreover, the differences in the amount of bulk shortening between models II and III allow a comparison of (i) magnetic fabric of inverted basins with that of the extensional stage, i.e. prior to shortening, (ii) basin development and inversion with the same amount of bulk extension and shortening, and (iii) magnetic fabric at normal faults and thrusts with similar displacement.

During deformation, the models were monitored by a series of photos from all sides. After the final stage of deformation, models were carefully wetted for vertical sectioning parallel to the extension and shortening direction, as well as for allowing model sampling for the AMS analysis (Fig. 1b). Each section was taken individually and had a width of 2–2.5 cm. Before sampling a section, the next section was prepared and a stable plate was placed between the two sections. This stable plate supported sampling of the outer section without exerting pressure on the rest of the model (i.e. where the next section would be taken; see Fig. 1b). During the sampling of a section, oriented plastic cubes (internal volume of 1.7 cm3) were carefully pressed horizontally into the cohesive material. Afterwards, the cohesive and wetted AMS samples were stored in semidry conditions (fridge with 7 ∘C and humidity of 75 %), allowing for AMS measurements over a few weeks before the material inside the plastic cube lost cohesion.

In total, 721 samples (Model I: 217, Model II: 241, Model III: 263) were taken across the models targeting the different structures. The focus of sampling was to acquire magnetic fabrics of different parts of the models (e.g. normal faults, thrusts, graben, footwalls, hanging wall blocks). This exercise eliminated the effect of measuring a bulk mixed AMS fabric, which may be created due to a small structure-to-sample-size ratio. However, it is not possible to entirely diminish this effect at faults. Sampling at faults covers the narrow fault zone and the vicinity of a fault. The vicinity of a fault might have a different magnetic fabric than observed directly at a fault plane. Consequently, a mixed fabric is represented by the bulk measurement of samples from fault zones. Such an effect needs to be considered during magnetic fabric interpretation of fault-associated datasets. Therefore, a structure-to-sample-size ratio is calculated (see Fig. S1 in the Supplement).

The bulk susceptibility and magnitude of AMS samples were measured with a MFK1-FA Kappabridge (Agico Inc.) in a low field (976 Hz) with an AC field strength of 200 A m-1 at room temperature. The grains within a sample respond to an applied external magnetic field, and the directional variation of the “response” (magnetic susceptibility) is described through a symmetric second-rank tensor with six independent matrix elements. The eigenvalues and eigenvectors of the matrix are the principal axes of susceptibility (kmax⁡≥kint≥kmin⁡) that describe the magnitude, orientation, and shape of a magnetic susceptibility ellipsoid. The maximum axis (kmax⁡) describes the magnetic lineation, whereas the plane containing the maximum and intermediate axes (kmax⁡ and kint) describes the magnetic foliation. Orientation of the principal axes was plotted in equal-area lower-hemisphere projections with north assigned towards the backstop of the model and the primitive circle (outer circle of projection) being parallel to the initial horizontal layering/bedding. The corrected degree of anisotropy, Pj, can be written as follows: 1Pj=exp⁡2nmax⁡-nmean2+nint-nmean2+nmin⁡-nmean2, where nmax⁡=ln⁡(kmax⁡), nint=ln⁡(kint), nmin⁡=ln⁡(kmin⁡), and nmean=(nmax⁡+nint+nmin⁡)/3. Pj reveals information about the sorting of grains within a sample, with a high degree of anisotropy corresponding to a preferred alignment of grains, whereas a low degree of anisotropy indicates a variation of grain orientations (Hrouda, 1982). Additionally, the shape of the susceptibility ellipsoid is described by T, 2T=2nint-nmax⁡-nmin⁡nmax⁡-nmin⁡, where T=1 represents an oblate shape, T=0 is a neutral triaxial shape, and T=-1 is a rotational prolate shape. The principal axes, the corrected degree of anisotropy (Pj), or the shape of anisotropy (T) were statistically interpreted and visualized with the help of graphical tools, MATLAB, and ArcGIS. The centre of each AMS sample defined the distance to a fault or the model surface (i.e. depth). Samples that were not perfectly located with their centres on a fault were still assigned to the fault-induced AMS dataset. Therefore, we introduced a threshold with a range of 0.8 cm (centre to corner of a sample) between their centre to a fault. Samples located within this threshold were labelled to be fault-induced.

In section view, the area of a sample was compared to the area of the fault zone, and a structure-to-sample-size ratio is calculated (Fig. S1). This ratio allows specifying the amount of the AMS signal induced by a fault relative to that induced by the unfaulted area within the sample. Moreover, this ratio can explain a broad scattering in the magnetic foliation (i.e. girdle distribution of kmax⁡ and kint axes) from data collected along the faults (Fig. S2).

Further uncertainties in the AMS datasets can be related to the sampling procedure, sample handling during measurements, and the instrument itself. The signal sensitivity of the instrument is 2 × 10-8 [SI], which is well below the signal of samples in this study. Using a relatively high content of ferromagnetic minerals produces a clear signal with narrow confidence ellipses and high F values (see dataset in Schöfisch, 2022). The F values provide information about the anisotropy of the measured material and show a relation between measured principal susceptibility axes and measurement errors (Jelínek, 1977). Confidence ellipses and measurement errors are not shown in the figures, as the symbols in the figures would overlap the uncertainty estimated from measurements. Additional variations in orientation in the principal axes can derive from sampling or by adjusting the samples in the instrument. In both cases, the sample can deviate from alignment with the reference/modelling north. However, the large amount of data from the different areas across the models provide a basis for statistical analysis and average out outliers in the dataset. Further scattering in the datasets/figures is addressed in the Discussion section. Note that interpretations of the magnetic fabric within the grabens are limited to the small number of samples and no solid statistical interpretation can be taken for internal graben changes.

Model I developed an east–west-striking graben bounded by two normal faults that dip 60–70∘ (Fig. 2). The northern fault (Normal Fault B) shows a displacement of ∼ 1.4 cm. With progressive extension, a couple of synthetic and antithetic faults form in the centre of the graben (Fig. 2). The AMS analysis of different parts of this model reveals that there is no sign of deformation in the footwalls on either side of the graben; they only show the oblate initial fabric produced by sieving, with kmin⁡ axes clustering vertically, as the pole to bedding, and kint and kmax⁡ axes spread in the horizontal plane along a primitive circle (Fig. 2). The principal axes in Footwall B show more scattered orientation and have wider confidence ellipses (Figs. 2 and S3). However, they display a comparable distribution in the degree of anisotropy similar to that observed in Footwall A (Fig. 3b). Within the graben, the magnetic fabric is similar to the fabric observed in the footwalls and shows the initial fabric (Fig. 2). Along the normal faults, kmax⁡ and kint axes form a sub-horizontal (10–20∘) magnetic foliation with the same dip direction as that of the normal faults; dipping north along Normal Faults A and south along Normal Fault B (Fig. 2). However, the magnetic foliation and inclination of the normal faults are oblique to each other (∼ 50∘). The kmin⁡ axes display rotation away from its vertical position into the opposite direction of the fault dip, as it is perpendicular to the magnetic foliation (Fig. 2). Moreover, the magnetic fabric is mostly oblate, but the degree of anisotropy is on average lower along the normal faults (1.14 < Pj < 1.40) compared to the fabric in the graben and its footwalls (1.15 < Pj < 1.51) (Fig. 3a and c).

The second model (Model II) was extended 1 cm before it was later shortened 1.5 cm (V1 in the Supplement). Similar to Model I, an east–west-striking graben developed during the extension phase (Fig. 4). With the onset of subsequent shortening (i.e. beginning of inversion) of the model, the normal faults steepened by ∼ 2–3∘, which led to a slight narrowing of the graben (Fig. S1). The final dip of the normal faults in this shortened model is steeper (70–85∘) than that of the normal faults of Model I, which underwent only an extension phase. However, the normal faults did not display any significant inversion during the subsequent shortening. Instead, precursors of gently dipping thrusts (25–35∘) developed as kinks, which offset footwall markers by a few millimetres. These thrusts are less mature than the normal faults as they developed less displacement. The thrusts divided footwalls A and B of the graben (Fig. 2) into footwalls and hanging walls (Fig. 4). Moreover, hanging walls A and B of the graben developed into a large pop-up structure (including the graben structure) that was uplifted along the thrusts during inversion. For comparison of the magnetic fabric and its development in the different areas, we labelled the different blocks based on their relation to the thrust faults (Footwall A, Hanging Wall A, etc.).

The magnetic fabric of footwalls A and B, as well as of the hanging wall blocks A and B, show an oblate magnetic fabric that is similar to the initial fabric with a vertical kmin⁡ axis and a horizontal spread of kmax⁡ and kint axes (Fig. 4). It is noted that the kmax⁡ distribution creates a subtle magnetic lineation parallel to the east–west axis in these areas away from the faults, and the confidence ellipses are narrower compared to the same areas in Model I. Furthermore, there is no clear distinction between the degree of anisotropy between the footwall (1.19 < Pj < 1.48) and hanging wall areas (1.17 < Pj < 1.45) of Model II (Fig. 3b). Additionally, the central graben reveals similar magnetic fabrics as those in footwall and hanging wall blocks (Fig. 4). The magnetic fabrics at the normal faults A and B display a distribution of subvertical kmin⁡ axes. The mean of the subvertical kmin⁡ axes is tilted opposite to the dip direction of the normal faults and points steeply to the south for Normal Faults A and to the north for Normal Fault B. The kmax⁡ and kint axes are mostly plunging gently (< 30∘) with a dominant east–west orientation for kmax⁡ axis and north–south for kint axis (Fig. 4). The principal axes are similarly clustered at the normal faults of Model II compared to Model I, in particular for the kmax⁡ orientations. However, the plane created by kmax⁡ and kmin⁡ axes (i.e. magnetic foliation) shows little to no inclination with regards to that along the normal faults in Model I. The AMS data from the thrusts in Model II show a distribution of principal axes similar to the normal faults of the same model (Fig. 4). However, the kmax⁡ and kint axes for each thrust tend to define a magnetic foliation subparallel to the thrusts. In Model II, the magnetic fabric along both the normal faults and the thrusts is mainly oblate with some occurrences of the prolate shape (Fig. 3c). The degree of anisotropy is comparable between the normal faults (1.12 < Pj < 1.38) and thrusts (1.14 < Pj < 1.41), but on average, it is lower than in areas away from the faults (Fig. 3b).

Model III shows similar structures to those in Model II (V2 in the Supplement). However, the thrusts in Model III, which was subjected to a larger amount of bulk shortening (+2.5 cm more than Model II), are more mature and display a larger displacement than those in Model II (Fig. 5). The normal faults dip 70–80∘ and show a displacement of ∼ 1.4 cm on average. During inversion, Normal Fault B steepened by ∼ 5∘, whereas Normal Faults A maintained the same dip as that prior to shortening (Fig. S1). Similar to Model II, with the subsequent shortening of Model III, the pre-existing graben narrowed with rotation of the normal faults by a few degrees (Fig. S1). Moreover, Hanging Wall A shows a minor block rotation (∼ 10∘) along Thrust A during the development of the main pop-up structure, which involves folding of the layers in the vicinity of the thrust. Both thrusts dip in a range between 25–40∘. However, Thrust A developed a splay at deeper parts of the model and shows less displacement than Thrust B due to a different accommodation of strain in this area (Fig. 5). Thrust B shows a displacement of ∼ 1.4 cm, which is similar to that along the normal faults. Similar to Model II, we divided and labelled the different blocks of Model III individually based on their relation to the thrusts (Footwall A, Hanging Wall A, etc.).

In Model III, the magnetic fabric in footwalls A and B, the hanging wall blocks A and B, and the graben developed similar magnetic fabrics as those observed in equivalent areas of the other two models (i.e. models I and II). However, there are minor deviations from the initial fabric in certain areas, particularly when comparing the clustering of the principal axes and their confidence ellipses. For example, Hanging Wall A developed a slight tilt in the magnetic foliation plane (∼ 10∘) that corresponds with the slight rotation of bedding during uplift of this block (Fig. 5). Furthermore, in footwalls A and B, and in Hanging Wall B, the kmax⁡ axes are clustering horizontally (i.e. creating a magnetic lineation) along east or west directions instead of spreading around the primitive circle. Furthermore, kint axes also cluster mainly in a north and/or south direction (Figs. 5 and 6).

The distribution of the principal axes (kmax⁡, kint, and kmin⁡ axes) along the normal faults in Model III is comparable to that along the normal faults of Model II (Fig. 6). The cluster of kmin⁡ axes (i.e. mean of kmin⁡ axes orientations) rotates slightly away from its vertical orientation to subvertical orientation, dipping south for the north-dipping Normal Faults A and dipping north for the south-dipping Normal Fault B. The AMS analysis does not show a clear girdle distribution of kmax⁡ and kint axes (i.e. magnetic foliation) parallel to the normal faults. Instead, kmax⁡ and kint axes are more clustered than the axes in models I and II (i.e. less stretched confidence ellipses), and the plane described by both axes (i.e. magnetic foliation) is almost not inclined at Normal Faults A and slightly inclined (∼ 10∘) at Normal Fault B (Figs. 5 and 6). However, kmax⁡ axes cluster horizontally towards the east or west (i.e. perpendicular to the extension and shortening directions), whereas kint axes distribute along a north to south axis (i.e. parallel to extension and shortening directions). Moreover, the magnetic foliation has similar orientation as the fault surfaces themselves, although both planes are oblique (50–60∘) to each other (Fig. 5). In contrast, the magnetic foliation is parallel to the thrust in Model III. Even though displacement along the thrusts is comparable to that along the normal faults, the magnetic foliation associated with thrusting is distinct (cf. Schöfisch et al., 2022). The degree of anisotropy of the normal faults (1.16 < Pj < 1.43) is distributed similarly to that of the thrusts (1.14 < Pj < 1.38) (Fig. 3d). The shape of anisotropy along the normal faults and thrusts plots mainly in the oblate field with some degree of prolate signature (Fig. 3c). The degree and shape of anisotropy along the structures in Model III are similar to that in the other two models (Fig. 3c). Moreover, the degree of anisotropy along the faults is on average lower than that observed in the footwalls (1.2 < Pj < 1.46) and hanging walls (1.18 < Pj < 1.47) (Fig. 3d).

The initial fabric of the models was created by sieving, where kmin⁡ axes cluster as the pole to bedding, and kmax⁡ and kint axes orient randomly in the horizontal plane parallel to bedding. This initial fabric is the reference and changes from this initial fabric are attributed to deformation. Sieving the initial magnetic fabric is a novelty, tested in this study. The sieved fabric differs from the scraped fabric; a scraped initial fabric shows horizontal alignments of kmax⁡ axes parallel and kint perpendicular to the scraping direction (Almqvist and Koyi, 2018; Schöfisch et al., 2020, 2022), whereas kmax⁡ and kint axes in a sieved magnetic fabric distribute randomly in different horizontal directions. Such a sieved magnetic fabric is similar to a sedimentary fabric that is observed in nature (cf. Borradaile and Henry, 1997; Bakhtari et al., 1998; Parés et al., 1999) and allows an improved interpretation and comparison between models and natural prototypes (e.g. Figs. 6 and S3).

The footwalls A and B of Model I are undeformed and still reveal the initial fabric after extension. It can be noted that the magnetic fabric in Footwall B of Model I has a relatively larger scatter compared to Footwall A of the same model (Figs. 2 and S3). Footwall B is resting and carried on the basal plate that moves during extension, whereas Hanging Wall A is resting on a stationary base. We assume that the deviation in magnetic fabric is due to grain reorientation/bulk compaction of sand due to vibration during movement of the underlying plate. An undeformed pile of sand is more prone to vibration than a pile that is under compression. Therefore, we assume that an influence of a vibration is more obvious during the phase of extension compared to shortening. The slight vibration could be because of a minor gap between the table and the metal plate. However, the weight of the model above the plate was removing this gap, but still, such a gap could be an explanation for a potential source of vibration. The modelling setup was reworked during the preparation of models II and III, and the gap between the table and metal plate was removed. Models II and III also retain the initial fabric in some locations even after model inversion, but both models also developed a clustering of principal axes with narrower confidence ellipses that is attributed to penetrative strain (Fig. 6). Consequently, the magnetic fabric in models II and III away from faults represent a mixed fabric between the initial and penetrative strain-induced fabric. These locations with initial fabric in models II and III indicate undeformed areas. Such undeformed areas during basin development are also known from natural examples (Oliva-Urcia et al., 2013; García-Lasanta et al., 2018).

In a natural sedimentary basin, a magnetic lineation (i.e. kmax⁡ clustering) develops parallel to the axis of extension. This observation derives from studying the reorientation of phyllosilicates in clay-rich sediments (e.g. Mattei et al., 1997, 1999; Cifelli et al., 2005). However, the surface markers of the models of this study indicate movement of the individual developed hanging wall and footwall (i.e. graben and Footwall B) without noticeable internal deformation (i.e. stretching) of the lithology during extension of the models. The magnetic fabric within the hanging wall and footwall of Model I has not developed a clear magnetic lineation parallel to stretching (Figs. 2 and S3). It rather indicates localized deformation, where the normal faults are developed (Fig. 2). The lack of pervasive extensional fabric in the model is likely a consequence of the granular material used in the model that has very low cohesion (cf. Eisenstadt and Withjack, 1995; Eisenstadt and Sims, 2005). However, our results underscore that there is minimal to no layer-parallel deformation/extension within the different hanging walls and footwalls during extension in such a brittle deformation environment (i.e. using a ridged velocity discontinuity/basal plate for initiating deformation).

The normal faults of Model I show a magnetic foliation that vaguely align parallel to the fault surface (note: normal faults of models II and III are discussed in Sect. 4.3.2). However, this alignment is oblique and there is a large difference of ∼ 40–50∘ between the inclinations of magnetic foliation and the fault surface (Figs. 2 and 6). Nevertheless, the dip direction of the magnetic foliation and fault surface are identical. Dilation is involved in the formation of normal faults in granular material. Dilation on the one hand forms a weak zone that is important during later basin inversion, but on the other hand, it is responsible for developing the observed magnetic fabric at the normal fault. The subangular non-cohesive grains rotate with their kmax⁡ axis along the normal faults into a clustering in the east or west direction that is parallel to the fault plane but perpendicular to the direction of model extension. The clustering is also reflected by the lower degree of anisotropy that is observed at the normal faults compared to the rest of Model I, where initial fabric dominates (Fig. 3a). The lower degree of anisotropy reflects a greater alignment of the magnetic grains at the normal faults compared to the grains away from the faults. However, the described oblate magnetic ellipsoid is not aligned with the fault surface (Fig. 3c), as the kint axes and, in general, the magnetic foliation are oblique to the fault plane (Figs. 2 and 6). It can be interpreted that the grains are sliding along the fault and rotate, with a tendency of being tilted along dip direction of the fault but not being aligned completely with the fault surface.

Overall, the magnetic fabric of the normal faults differs in the degree of anisotropy and clustering of principal axes to the initial fabric that is observed in the rest of the model. These differences in magnetic fabric in Model I are results of extensional deformation. Although there is no presence of a layer-parallel extensional fabric, an “extensional fault-induced fabric” (i.e. normal fault-induced fabric) developed in Model I as consequence of localized deformation during basin development.

Applying AMS allows visualizing deformation in sandbox models (Almqvist and Koyi, 2018; Schöfisch et al., 2020, 2022). In the models of this study, the magnetic fabric also reflects deformation and the development of structures. In addition, this study reveals an overprint of magnetic fabric due to inversion; specifically, differences are monitored between extensional to compressional tectonic environments. Extension did not result in penetrative strain in the models, which is indicated by a persistent initial fabric throughout the hanging wall (i.e. graben) and footwalls of Model I as well as the sharp change in the degree of anisotropy between normal faults and in areas away from the faults. In contrast, in shortened models (e.g. Schöfisch et al., 2022) or inverted models (models II and III of this study), shortening leads to the development of penetrative strain in areas away from the faults. Consequently, the magnetic fabric is sensitive to strain changes in compressional regimes, but further studies are required for depicting extensional fabrics in more detail in sandbox models.

As the models simulate brittle behaviour of upper crustal rocks without taking into account processes like crystal–plastic deformation, fluid migration, and recrystallization of magnetic contributors (i.e. changes in magnetic mineralogy and development of sub-fabrics), changes to the initial fabric in the models are solely related to grain reorientation. Such a modelling setup and the combination of sandbox modelling with magnetic fabric analyses enables investigation, visualization, and highlighting the importance of grain reorientation in natural analogues.

As Eisenstadt and Sims (2005) and Deng et al. (2019) documented, there is no or very limited reactivation of pre-existing normal faults during the inversion of such a model setup using loose sand above a ridged basal plate. In addition, an extensional fabric away from normal faults as observed in nature (e.g. Sagnotti et al., 1994; Mattei et al., 1997; Borradaile and Hamilton, 2004; Cifelli et al., 2005) is not displayed in these models. Therefore, it may be necessary to prepare similar experiments simulating the development of a basin and its inversion with higher complexity, e.g. testing syntectonic basin sedimentation to create magnetic lineation in basin fill or using different materials (e.g. wet clay) to produce extensional structures (e.g. roll-over anticlines; cf. Eisenstadt and Withjack, 1995; Eisenstadt and Sims, 2005). Moreover, different materials (Eisenstadt and Sims, 2005), oblique inversion (Nalpas et al., 1995; Brun and Nalpas, 1996; Dubois et al., 2002; Deng et al., 2019), or different modelling setups with viscous décollement (e.g. Roca et al., 2006; Del Ventisette et al., 2006, and references therein) lead to a reactivation of normal faults within the models. In such cases, investigating a magnetic fabric overprint due to fault reactivation is of great interest.