Insights into the Paleostress Analysis of Heterogeneous Fault-Slip Data by Comparing Different Methodologies: The Case of the Voltri Massif in the Ligurian Alps (NW Italy)

Abstract

One of the most critical stages in fault-slip data stress analysis is separating the fault data into homogeneous subsets and selecting a suitable analysis method for each subset. A basic assumption in stress tensor computations is that fault activations occur simultaneously under a homogeneous stress regime. With that rationale, this work aims to attain improvements in the paleostress reconstruction from the polyphase deformed region of Voltri Massif in the Ligurian Alps by using already published heterogeneous fault-slip data inverted using best-fit stress inversion methods and in the absence of any tectonostratigraphic and overprinting criteria. The fault-slip data are re-examined and analyzed with a best-fit stress inversion method and the Tensor Ratio Method (TRM) in the absence of any tectonostratigraphic and overprinting criteria. This analysis defines crucial differences in the paleostress history of the Voltri Massif in the Ligurian Alps, and gives insight into the analysis and results of different stress inversion methodologies. Best-fit site stress tensors have substantial diversity in stress orientations and ratios, implying possible stress perturbations in the region. The reason for these diversities is that the Misfit Angle (MA) minimization criterion taken into account in the best-fit stress inversion methods allows for acceptable fault-slip data combinations, which under the additional geological compatibility criteria used by the TRM, are found to be incompatible. The TRM application on this already published and analyzed data defines similar site and bulk stress tensors with fewer diversities in stress orientations and ratios defined from fault-slip data whose orientations always satisfy the same additional geological compatibility criteria induced by the TRM, and not only from the MA minimization criterion. Thus, TRM seems to define stress tensors that are not as sensitive to the input of fault-slip data, compared to the best-fit stress tensors that appear to suffer from the ‘overfitting’ modeling error. Five distinct TRM bulk paleostress tensors provide a more constrained paleostress history for the Voltri Massif and the Ligurian Alps, which after the restoration of the ~50° CCW rotation, comprise: (a) a transpression–strike-slip stress regime (T1) with NNE-SSW contraction in Late Eocene, (b) an Oligocene NW-SE extensional regime (T2), which fits with the NW-SE extension documented for the broader area north of Corsica due to a significant change in subduction dynamics, (c) a transient, local, or ephemeral NE-SW transtension (T3) which might be considered a local mutual permutation of the T2 stresses, and (d) a Miocene transpression with a contraction that progressively shifted from ENE-WSW (T4) to NNE-SSW (T5), reflecting the stress reorganization in the Ligurian Alps due to a decrease in the retreating rate of the northern Apennines slab. Therefore, paleostress reconstruction can be fairly described by enhanced Andersonian bulk stress tensors, and requires additional geological compatibility criteria than the criteria and sophisticated tools used by the best-fit stress inversion methods for separating the fault-slip data to different faulting events.

1. Introduction

The ‘inverse problem’, i.e., finding the driving best-fit stress tensor of a given fault population, has been central to the geological community for at least 40 years. Only four parameters, i.e., the orientations of the three principal stress axes (σ₁, σ₂, and σ₃) and the stress ellipsoid shape ratio R = (σ₂−σ₃)/(σ₁−σ₃) with 0 ≤ R ≤ 1 [1,2] are needed to obtain the driving stress tensor. Using numerous stress inversion algorithms with the aid of available software can quickly solve the inverse problem, e.g., [1,2,3,4,5,6,7,8,9,10,11,12]. Despite the debate on whether the determined principal stress axes represent strain (kinematics) or stress (dynamics) (cf. [13]), there is an increasing number of methods for interpreting fault-slip data and calculating the driving stress tensors. The solutions are mathematically robust and not time-consuming for best-fit stress inversion methods, since they require at least four differently oriented fault planes with slickenlines [1,14]. Nonetheless, they are geologically admissible only if the following fundamental assumptions are satisfied, e.g., [2,3]: (1) The fault-slip data are homogeneous, i.e., they have been activated simultaneously under the same regional stress regime, (2) the orientation of the fault planes is random, (3) displacements on the fault planes are small concerning their lengths, (4) there are no rotations of the fault planes, and (5) slips on the fault planes are independent, and therefore, there is no fault interaction.

Best-fit stress inversion algorithms, e.g., [3,5], were deployed to define the four parameters of the driving stress tensor without having any limitation in the orientation of the stress axes. However, they have been based on the analysis of [15], which is carried out with a stress tensor, having one of its principal stress axes in a vertical position, like the Andersonian model [16]. More precisely, [15] evidenced that any fault slip, even an oblique-slip one, coinciding with the maximum shear stress, i.e., the Wallace-Bott hypothesis [15,17], can be driven by a stress state with a vertical principal stress axis depending only on the stress ratio R.

Despite some arguments about the validity of the Wallace-Bott hypothesis (e.g., [18,19,20,21], the ‘enhanced’ Andersonian model with the addition of the stress ratio succeeds in describing any fault activation, even those driven by transpressional (TRP) or transtensional (TRN) stress regimes, without the need for the principal stress directions to rotate away from the recommended positions of [16],[22,23]. Moreover, [24,25,26] presented a theoretical analysis, i.e., the Slip Preference Analysis (SPA), which indicates the geometry and kinematics of all possible optimal fault planes and their activations under the different enhanced Andersonian states of stress, i.e., extension, compression, and strike-slip, enhanced with the stress ratio values. The importance of the Slip Preference Analysis is that it provides additional compatibility criteria for the simultaneously activated faults under the enhanced Andersonian stress regimes, and therefore, the homogeneity of the fault-slip data.

Geoscientists have tried to unravel the paleostress history of many regions by defining the ‘mean’ or ‘average’ stress tensors that can better describe and represent the different tectonic events and concomitant fault reactivations over the rock volume investigated (e.g., [27,28]). They use fault-slip data with various orientations recorded at different sites by applying several best-fit stress inversion methods [29,30,31,32,33].

The best-fit stress methods define stress tensors based only on the Misfit Angle (MA) minimization criterion between the observed fault slickenline and slip preference (SP) [24]. Slip preference is a term that refers to the expected (theoretical) fault slip under a stress regime, and in the best-fit stress inversion methods, it is assumed to coincide with the direction and the sense of the maximum resolved shear stress (the Wallace-Bott hypothesis [15,17]). In general, if the MA of each activated fault is less than 20° ([1,34,35] and references therein), the stress tensor solutions are considered satisfactory, though the threshold angle influences the final results of the methods [36].

However, an individual fault might exhibit a polyphase activity in regions with complex tectonic history. As a result, the recorded fault-slip data are heterogeneous (or polyphase). In such cases, it can be quite puzzling to define exactly which faults have co-functioned under the different driving stress regimes with only the MA minimization criterion. This problem remains unsolved when using either the separation of a fault/slip dataset into independent subsets (hard division) [37,38,39] or not (soft division) [40]. As a result, in the presence of heterogeneous fault-slip data, the estimation of the optimal stress through the application of the best-fit stress inversion methods is difficult to be reached [38] because such solutions might be misleading due to the influence of faults belonging to other phases [36,38]. Such an estimation can be even more precarious, especially when stress tensors are calculated from < 10 fault-slip data, since such solutions can be found even from randomly generated faults [41]. Although the heterogeneous fault-slip data should be classified into homogeneous groups based on field evidence, this is hard to establish in most cases, like the Voltri Massif of the Ligurian Alps. Therefore, additional compatibility criteria based on the relationship between the fault activation and the driving stress regime are needed to separate the fault-slip data into homogeneous groups.

Regardless of the above, the need to define the stresses drifts scientists to analyze their fault-slip data with the available open-source software, making the process and the used software progressively widely accepted. Such widely used best-fit stress inversion methods are provided by the FSA software ([42] and subsequent modifications), which uses the Monte Carlo search method [1] and Win-Tensor software [43]. The process is iterative and repeats until the remaining data cannot define a physically meaningful stress tensor. Several published articles are based on this inversion approach without using tectonostratigraphic criteria, e.g., [29,32].

The study in [32] examined the polyphase brittle deformation in Voltri Massif and suggested a paleostress history using the FSA software [42] to separate heterogeneous fault-slip data into homogeneous groups, and the Win-Tensor software [43] for verifying this grouping. Both software use best-fit stress inversion algorithms. On the other hand, the study in [44], based not only on the MA minimization criterion of the best-fit stress inversion method, but also on additional compatibility criteria induced by the Slip Preference Analysis concerning the fault activations, developed a new separation and stress inversion method, the Tensor Ratio Method (TRM). This fact motivated us to test and examine whether additional compatibility criteria could assist with data separation and provide better resolved stress tensors that can enhance our understanding of polydeformed regions as the Voltri study area.

The purpose of this study is to examine: (1) whether the results concerning the fault activations in the region and the paleostress history deviate or not from that of [32], (2) the efficiency of the differently applied stress inversion methods and the usefulness of implementing different constraining compatibility criteria among the fault-slip data, and (3) a better understanding of the stress regimes as provided by the site and bulk resolved stress tensors.

2. Geological Setting

The Western Mediterranean region (Figure 1a) has experienced a complex tectonic history since the Mesozoic-Cenozoic, involving the opening and destruction of the Late Triassic/Jurassic Ligurian Ocean and west Alpine Tethys through curved-shaped subduction zones of various polarities [45,46]. The Voltri Massif in the Ligurian Alps (Figure 1a) is an HP eclogite-bearing ophiolite domain consisting of a tectonically complex area between the Western Alps and the Northern Apennines. The Western Alps are characterized by a main westward tectonic nappe-stacking [47,48,49], and the Northern Apennines, by an east/northeastward vergent fold-and-thrust belt [50,51].

In terms of the tectonostratigraphic framework, the Voltri Massif consists of three main tectonic units [52,53]. These units (Figure 1b), i.e., the Erro-Tobio serpentinized lherzolite, the Beigua serpentinite, and the Piedmontese Nappe, are derived from (a) oceanic crust and mantle, (b) a continental margin, and (c) flysch units derived from the sedimentary cover of an oceanic basement [53]. The boundary between the Western Alps and Northern Apennines is marked by the Sestri-Voltaggio Zone [54], which lies east of the Voltri Massif (Figure 1b) and separates the ophiolitic domain of the Ligurian Alps from the Northern Apennines. This zone is 5–6 km wide, forming an N-S strip from Sestri Ponente in the Ligurian Sea up to Voltaggio in the north [55] (Figure 1b), and it is characterized by complex kinematics related to the main phases of the Alpine orogenesis [56]. It is also referred to as a Sestri-Voltaggio Fault [57] since it records Oligo–Miocene brittle kinematics [58,59].

Figure 1. Geographical and geological information concerning the study area: (a) Satellite image of the Western-Central Mediterranean (modified from [57] and references therein). Explanation: LA: Ligurian Alps, VM: Voltri Massif, WA: Western Alps. (b) Simplified geological map of the Voltri Massif and location of the structural stations (modified after [32,53,55,60]). Explanation: the numbered red squares are the structural stations (SS) with fault-slip data, as presented by [32], SVF: Sestri-Voltaggio Fault.

Figure 1. Geographical and geological information concerning the study area: (a) Satellite image of the Western-Central Mediterranean (modified from [57] and references therein). Explanation: LA: Ligurian Alps, VM: Voltri Massif, WA: Western Alps. (b) Simplified geological map of the Voltri Massif and location of the structural stations (modified after [32,53,55,60]). Explanation: the numbered red squares are the structural stations (SS) with fault-slip data, as presented by [32], SVF: Sestri-Voltaggio Fault.

The present structure of the Apennines resulted from the long-lasting interaction between the African plate or its indenter (i.e., Adriatic-Apulia foreland) and the European plate (Corsica-Sardinia foreland). This interaction involved processes like a Triassic to Early Cretaceous rifting and oceanic spreading, transtension and opening of a young oceanic basin of the Tethyan margin (i.e., the Ligurian-Piedmont ocean), as well as a Late Cretaceous–Early Tertiary eastward intraoceanic subduction that changed polarity to westward subduction in the Paleocene (from Alpine to Apennine) at the European-Adria plate margin [61,62,63,64].

Due to the structural position of the Voltri Massif between the two orogens, the Tertiary late-orogenic processes have been described and interpreted differently amongst geoscientists, and the paleostress history remains under debate. This complex scenario is also shown in the evolution of the Tertiary Piedmont Basin (TPB), an Oligo–Miocene wedge-top basin located next to the Voltri Massif and between the two opposite verging orogens (Figure 1). The sedimentation in the TPB occurred in three main tectonic episodes: the exhumation of the Ligurian sector of the Western Alps, the opening of the Liguro-Provençal basin, and the formation of the Apennines thrust belt [57]. In particular, various data show that the brittle-ductile to brittle deformation of both the basement rocks and the TPB deposits was very complex from the Oligocene onwards, since folding and thrusting, as well as normal and strike-slip faulting, were documented, e.g., [58,65,66,67,68].

Tectonic reconstructions have interpreted the Voltri Massif as an extensional domain, accommodating lithospheric thinning since the Late Eocene–Early Oligocene [69], or as an exhumed terrain driven by means of polyphase compressional structures [70,71]. The study in [58] attributed the latter to a transpressive regime triggered by the Corsica–Sardinia counterclockwise rotation [69].

In the absence of stratigraphic constraints, the paleostresses and the fault evolution are challenging to unravel. Basement faults (i.e., thrusts) in the Ligurian Alps and North Apennines were active during the Oligo–Miocene and unsealed by the Oligocene transgression [72]. As a result, several tectonic events have been characterized by temporal overlapping and the reactivation of pre-existing structures.

Although detailed studies were carried out in the eastern part of the study area, close to the Sestri Voltaggio Zone (east of the village of Rossiglione; Figure 1b), these provide different conclusions showing the complexity of the brittle-ductile to brittle deformation of the Voltri Massif. In particular, one group of authors [60,73,74] describes the main faults as subvertical, striking N-S, or NW-SE, with strike-slip to oblique-slip kinematics, defining a regional-scale dextral Riedel system, with associated top to the N-NE thrust faults, in the framework of a regional NE-SW trending maximum shortening. The second group describes mainly E-W and N-S or NNW-SSE striking faults, defining pull-apart basins filled with Pliocene sediments during a transtensional stress regime ([75] and references therein).

The study in [60] relates the Voltri Massif with the westward migration of the Alpine thrust front and the contemporaneous eastward retreat of the Apennine slab [76] during the Oligocene-Miocene times. According to [59], two major tectonic complexes exist in the Voltri Massif, separated by a major ductile-to-brittle multiple extensional detachment system (Figure 1b).

In contrast, [71] explains the deformation of the Voltri Massif and the Ligurian Alps with the back-thrust of the Ligurian Alps onto the North Apennine Units. The overall deformation is transpressive in the Oligocene–Miocene times triggered by the Corsica–Sardinia counterclockwise rotation. This transpressive deformation was accommodated by a complex dextral Riedel-type strike-slip fault system, which heavily segmented the Tertiary Piedmont Basin, affecting the sedimentation and the clast provenance of the Tertiary conglomerates.

This complex area of the Voltri Massif where stratigraphic constraints are lacking is used by [32] as a case study to unravel the polyphase brittle tectonics via different stress inversion software on heterogeneous fault-slip data in an area of about 20 km². They suggested the following paleostress history of the region since the Oligocene (see Figure 10 of [32]): (a) a Rupelian–Early Chattian (?) deformation event A, described by a strike-slip tensor with σ₁ trending NNW-SSE, (b) an Aquitanian–Early Burdigalian (?) deformation event B described by a strike-slip tensor with σ₁ trending NE-SW, and (c) a deformation event C described by an extensional/transtensional tensor with σ₃ trending either NW-SE or NE-SW, dated in Pliocene-Quaternary (?).

3. Methods and Fault-Slip Data

Among the several published suggested methods for the stress inversion of fault-slip data (see [23,27,44]), the best-fit stress inversions are still the traditional and most popular in dealing with real data, possibly because of the availability of the open-source software and the fast calculating process itself.

3.1. FSA and Win-Tensor

The driving best-fit stress tensors, determined by [32], were calculated with the aid of the FSA software ([42], and subsequent modifications) and the Win-Tensor software [43], by taking into account MA ≤ 30° in SSs shown in Figure 1b. The FSA software first calculates many reduced stress tensors through a random grid search, following the Monte Carlo search method of [1]. The Win-Tensor software initially estimates the parameters of a reduced stress tensor through an improved version of the Right Dihedra method. Afterwards, it defines the optimal stress tensor through an iterative rotational stress optimization process that further minimizes the slip deviations and favors slip on the fault planes [77].

Regarding the application of the Win-Tensor software on their database, the parameters for assessing the stress state’s quality are the same as the FSA software, i.e., the MA and its distribution (ideally unimodal) and the high-shear/normal stress ratio. We have to mention that the ratio of shear stress (τ) to normal stress (σ_n) on the fault plane is critical for its (re)activation, and it is called slip tendency T_s [78].

3.2. The Tensor Ratio Method (TRM)

The TRM, as described in [24,25,44], is a simple graphical and semi-automatic method plotting the heterogeneous fault-slip data on TR diagrams and examining if the slips on the faults are simultaneously compatible with a specific orientation of the Andersonian stress axes and a specific stress ratio (R) value (for details, see [44]). Moreover, it can define possible spatial stress perturbations due to large fault structures, as shown in the case of the 1999 Chi-Chi earthquake [25]. The faults with slips characterized by TR compatibility are called Tensor Ratio Compatible Faults (TRCF), defining a specific stress ratio, i.e., the stress ratio of the enhanced Andersonian driving stress tensor. The optimal TRM stress tensor is the one that explains the largest number of the TRCF with MA ≤ 20°, and secondly, their Mean Misfit Angle (MMA), which must be the smallest. Although in TRM, the default threshold angle for the MA is 20°; in the present examination, for the sake of consistency, we elaborated the optimal TRM stress tensors using MA ≤ 30°, i.e., the same threshold angle as that chosen by [32].

The TRM uses the MA minimization criterion and additional compatibility criteria concerning the fault activation, as the latter resulted from the Slip Preference Analysis [24,25,26,44]. More precisely, it has been shown that: (1) extensional and compressional enhanced Andersonian stress regimes could activate, respectively, only extensional and contractional faults; (2) a contractional or an extensional fault can be activated if only its slickenline is in the side where the fault dip direction has an acute angle with the horizontal σ₁ or σ₃ principal stress axis, respectively; (3) faults with dip directions at angles up to ±15° with the horizontal principal stress axis, either σ₁ or σ₃, activate as dip-slip faults (either reverse or normal), and these faults have the highest slip tendency values; (4) the (sub)horizontal kinematic axes (either P or T) of the faults with pitch (pt) ≥ 80°, trend along the horizontal σ₁ or σ₃ axis, and the faults with pt ≥ 60° trend very closely around it; and (5) enhanced Andersonian stress regimes, either extensional or compressional, of similar orientations but of different stress ratios activate different Tensor Ratio Compatible Faults with Slip Preferences outlining different plot regions on the TR diagrams.

3.3. Fault Slip-Data

In the present study, fault-slip data (Appendix A) used by [32] for defining the paleostress history in the Voltri Massif of the Ligurian Alps (Figure 1a) are re-analyzed for un-raveling the paleostress history through a different approach. They are 92 fault-slip data in total, and they have been recorded in five structural stations (SS), i.e., SS2, SS3, SS6, SS7, and SS11 from Pra’ Vallarino to Tiglieta. These SSs are located along an ENE-WSW to NE-SW boundary that separates the overlying Erro-Tobbio Unit in the north from the underlying Beigua Unit in the south (Figure 1b, and SS location details in Figure 2 of [32]). Both units belong to the Voltri Massif, and according to [32], there is no big fault in this area like the Sestri-Voltaggio fault (SVF) (see Figure 1 and Figure 2 in [32]). In contrast, [60] suggested that the contact is a large detachment fault (Figure 1b). From the 92 fault-slip data, only one that dips at less than 50° towards NNW with extensional displacement (FED_1, Appendix A) at SS2 might be similar to the detachment geometry and kinematics.

Fifty-one (51) fault-slip data are extensional, and 41 are contractional, based on whether the slip along the fault surface is uplifting or subsiding the hanging wall against the footwall block, and considering the horizontal plane as a reference level. Interestingly, 56 fault surfaces display pitch (pt) ≤ 30°, 22 have 30° < pt < 60°, and 14 have pt ≥ 60°, indicating the predominance of the strike-slip and oblique-slip fault motions over the dip-slip, which in turn does not favor the precise estimation of the least or greatest principal stress axis of the resolved stress tensors. We follow the stress types classification for the stress regimes of [79].

Finally, we chose these data for analysis and applied a different approach by defining both ‘site’ stress tensors at each SS and ‘bulk’ stress tensors from the whole fault-slip dataset to compare our results with their results. Furthermore, the two most remote SSs are at a distance of no more than 10 km from each other (Figure 1, and SS location details in Figure 2 of [32]), describing a rock volume that is small compared to the known large structures of the region like the SVF, and far apart from them, so that it can be considered as a small cubical element for the regional stress regimes [18,80].

In this study, the present analysis includes six stages, starting by examining the possible heterogeneity of the fault-slip data [32] at each SS (Figure 1b) by applying another best-fit stress inversion method, i.e., the ‘Minimized Shear Stress Variation,’ which uses the algorithm of [81,82] with the aid of MyFault software (

Table 1. Best-fit stress tensors with MyFault, software, and MA ≤ 30°.

ST	n	FTC	FTE	σ1	σ2	σ3	R	MMA	FT (MA ≤ 30°)	FT/n (%)	N	N (MA ≥ 30°)	SS
T_MFSS2	16	12	4	072°–30°	275°–58°	168°–11°	0.48	28.3°	14	88	1–16	2, 14	SS2
T_MFSS2F	14	12	2	065°–22°	293°–60°	164°–20°	0.68	7.0°	14	100			SS2
T_MFSS3	21	7	14	268°–78°	082°–12°	173°–01°	0.57	63.8°	6	29	17–37	18,19, 20, 21, 22, 23, 24, 27, 28, 29, 30, 32, 35, 36, 37	SS3
T_MFSS3F	6	0	6	274°–84°	081°–05°	171°–01°	0.45	9.8°	6	100		17, 25, 26, 31, 33, 34	SS3
T_MFSS6	13	8	5	156°–12°	036°–67°	250°–20°	0.39	63.3°	6	46	38–50	38, 39, 40, 42, 47, 48, 50	SS6
T_MFSS6F	6	5	1	154°–04°	052°–72°	245°–17°	0.47	7.1°	6	100			SS6
T_MFSS7	32	10	22	258°–57°	151°–11°	055°–31°	0.11	76.8°	5	16	51–82	51, 52, 53, 54, 55, 56, 58, 59, 60, 61. 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 74, 76, 77, 78, 79, 81, 82	SS7
T_MFSS7F	5	1	4	258°–52°	351°–02°	083°–38°	0.39	10.0°	5	100			SS7
T_MFSS11	10	4	6	009°–08°	266°–57°	104°–31°	0.53	62.7°	2	20	83–92	83, 84, 85, 86, 87, 90, 91, 92	SS11
T_MFSS11F	4	0	4	150°–44°	294°–40°	041°–19°	0.20	3.8°	4	100			SS11

The resolved best-fit site stress tensors at each structural station (SS), e.g., T_MF_SS2, with the use of the best-fit stress inversion method, “Minimized Shear Stress Variation” that uses the algorithm of [81,82]. Explanation: T_MF_SS2: Tensor (T), MF (MyFault), SS2: structural station 2, ST: Stress tensor, n: number of fault-slip data in the SS, FTC: contractional fault-slip data, FTE: extensional fault-slip data, R: stress ratio, FT (MA ≤ 30°): Fault-slip data with Misfit Angle ≤ 30°, MMA: Mean Misfit Angle, FT/n: percentage of the fault-slip data that have MA ≤ 30° out of the total number (n), N: The number of the fault-slip datum as shown in Table A1.

). The authors in [81,82], in estimating the regional stresses, made the simplifying assumption that the magnitude of the slip stress on the fault is similar for all faults in the set at the time of slip. Thus, minimizing the variations in slip stress among the faults leads to an overdetermined set of linear equations. These equations are solved using the standard eigenvector method, giving the three principal stresses and their direction. Because the mean stress during the slip is generally unknown, the principal stresses are normalized, assuming that the maximum stress is 1 and the minimum is 0. The stress ratio is equal to intermediate stress. The fault-slip data for which the obtained MA is greater than 30° under the resolved stress tensor were excluded at each station. However, if any of the excluded fault-slip data under the new resolved stress tensor obtain a new MA that is smaller or equal to 30°, they are reconsidered in the final solution.

4. Paleostress Analysis Results

Table 1 describes the resolved best-fit site stress tensors with MyFault software, verifying the fault-slip data’s heterogeneity as stated by [32]. Indeed, all SSs apart from the SS2 include heterogeneous fault-slip data, since the first resolved stress tensors, e.g., T_MF_SS3, and T_MF_SS7, explain a low percentage of fault-slip data with MA ≤ 30°, and the mean Misfit Angle (MMA) is much higher than 20° [83,84]. In SS2, however, the input fault-slip data’s heterogeneity shown by the MMA = 28.3° is not real, but is due to the fault-slip data with numbers 2 and 14 (see Appendix A). The latter describes slip vectors with opposite sense of slip, i.e., normal dextral (ND) in contrast to inverse-sinistral (IS) of the others (see Figure 8 in [32]). Excluding the incompatible fault-slip data, i.e., those with MA> 30°, five resolved stress tensors labeled with F, e.g., T_MF_SS7F (Table 1), can be accepted as candidate solutions since they explain all the input fault-slip data with MA ≤ 30°. However, all these solutions T_MF_SS*F have been defined with less than 10 input fault-slip data, apart from the T_MF_SS2F. In addition, three out of five, i.e., T_MF_SS2F, T_MF_SS7F and T_MF_SS11F, do not obey the enhanced Andersonian model, implying that the region was subjected to non-Andersonian stress states. The second stage refers to the TRM application on the fault-slip data of each SS for defining the enhanced Andersonian site stress tensors, as shown in

Table 2. The enhanced Andersonian site stress tensors (MA ≤ 30°) with the TRM application [41].

ST	N	FTE	FTC	TRCF	σ1	σ2	σ3	R	ST REG	MMA (MA ≤ 30°)	N (Appendix A)	SS
TSS2	16	4	12	11	072°–00°	072°–90°	162°–00°	0.01	TRP-SS	11.1°	3, 4, 5, 7, 8, 9, 10, 12, 13, 15, 16	SS2
T1SS3	21	14	7	4	014°–00°	014°–90°	104°–00°	0.05	TRP	9.0°	21, 22, 29, 35	SS3
T2SS3	21	14	7	6	047°–90°	137°–00°	047°–00°	0.91	TRN	15.0°	20, 27, 30, 32, 34, 37	SS3
T3SS3	21	14	7	5	346°–90°	076°–00°	346°–00°	0.95	TRN	10.6°	17, 23, 28, 31, 33	SS3
TSS6	13	5	8	6	159°–00°	159°–90°	069°–00°	0.67	SS-TRN	12.8°	(40), (41), 43, 44, 45, 46, 48, 49	SS6
T1SS7	32	22	10	5	046°–00°	136°–00°	046°–90°	0	TRP	9.4°	53, 57, 58, 62, 64	SS7
T2SS7	32	22	10	7	166°–90°	076°–00°	166°–00°	0.91	TRN	14.0°	60, 63, 68, 75, 76, 80, 82	SS7
T3SS7	32	22	10	8	103°–90°	013°–00°	103°–00°	0.96	TRN	15.5°	52, 54, 59, 74, 76, 78, 79, 80	SS7

The enhanced Andersonian site stress tensors, as defined at the different structural stations (SS) with the TRM application [44]. Explanation as in Table 1, TRCF: Tensor Ratio Compatible faults, ST REG: stress regime, N: extensional fault-slip data compatible with the TRM stress tensor. TRP: Transpression, SS: (pure) Strike-Slip, TRN: Transtension.

and Figure 2. In four out of five SS, enhanced Andersonian site stress tensors have been obtained (Table 2). In addition, in two SS, i.e., the SS3 and SS7, where more than 20 fault-slip data were recorded, site stress tensors of more than one were calculated. Most of the site stress tensors (except the SS2) were found with less than 10 fault-slip data, as the stress tensors defined by [32] and the ‘Minimized Shear Stress Variation’ best-fit stress inversion method. Such site stress tensors should be treated with great caution, since the MMA cannot be used as a quality indicator [41,83,84,85].

The third stage of the analysis includes the calculation of the bulk stress tensors by applying the TRM to the whole fault-slip dataset (Appendix A). The calculation of the bulk stress tensors eliminates possible stress heterogeneities due to the fault interaction or the existence of large structures (see also [29,30,85]). In particular, five bulk stress tensors, labeled T1_TRM through T5_TRM, have been calculated (Figure 3,

Table 3. The enhanced Andersonian bulk stress tensors with the TRM application [41].

ST	N	FTE	FTC	TRCF	σ1	σ2	σ3	R	ST REG	MMA (MA ≤ 30°)	N (Appendix A)	SS
T1TRM	92	51	41	8	143°–00°	143°–90°	053°–00°	0.18	TRP-SS	13.0°	43, 44, 45, 46, 69, 72, 77, 87	SS6 (4), SS7 (3), SS11 (1)
T2TRM	92	51	41	14	103°–90°	013°–00°	103°–00°	0.60	PE	13.4°	14, 25, 26, 40, 52, 54, 59, 70, 74, 80, 86, 89, 90, 91	SS2 (1), SS3 (2), SS6 (1), SS7 (6), SS11 (4)
T3TRM	92	51	41	14	175°–90°	085°–00°	175°–00°	1.00	TRN	9.6°	1, 17, 23, 28, 31, 33, 39, 60, 63, 65, 68, 73, 75, 82	SS2 (1), SS3 (5), SS6 (1), SS7 (7)
T4TRM	92	51	41	21	063°–00°	063°–90°	153°–00°	0.20	TRP-SS	11.4°	3, 4, 5, 7, 8, 9, 10, 12, 13, 15, 16, 18, 36, 42, 57, 58, 62, 64, 83, 84, 85	SS2 (11), SS3 (2), SS6 (1), SS7 (4), SS11 (3)
T5TRM	92	51	41	12	019°–00°	109°–00°	019°–90°	0.00	TRP	13.1°	21, 22, 29, 35, 47, 48, 49, 53, 58, 61, 85, 87	SS3 (4), SS6 (3), SS7 (3), SS11 (2)

The enhanced Andersonian bulk stress tensors, as defined at the different structural stations (SS) with the TRM application [44]. Explanation as in Table 1 and Table 2; fault-slip data ≥ 4 are shown in bold as the SSs, where they were recorded. TRP: Transpression, SS: (pure) Strike-Slip, TRN: Transtension. PE: Pure Extension.

).

T1_TRM bulk stress tensor is defined by only eight fault-slip data. It explains four fault-slip data in site SS6 and less than four in sites SS7 and SS11. It could be correlated with the T_SS6 site stress tensor, although a change in the trend of the horizontal stress axis is observed.

T2_TRM bulk stress tensor is defined by 14 fault-slip data and explains six fault activations in SS7, four at SS11, and less than four in the other SS.

T3_TRM bulk stress tensor is defined by 14 fault-slip data and explains five fault activations in SS3 and seven in SS7. It presents substantial similarity with the site stress tensors T3_SS3 and T2_SS7.

T4_TRM bulk stress tensor is well defined by 21 fault-slip data and explains 10 fault activations in SS2 and four in SS7, having substantial similarity with the site stress tensors T_SS2 and T1_SS7. It also explains less than four fault activations in other sites like SS11.

T5_TRM bulk stress tensor is defined by 12 fault-slip data and explains four fault activations in SS3 and less than four in SS6, SS7, and SS11. It presents a strong similarity with the T1_SS3 site stress tensor.

Since our interest is to compare the TRM with the best-fit stress inversion methods, a fourth analysis stage is performed. In this stage, we examine which additional fault-slip data (Appendix A) are compatible with the enhanced Andersonian bulk stress tensors of the previous stage if we only consider the MA minimization criterion of 30°, i.e., the constraint of the best-fit stress inversion methods used by [32]. All the resolved Andersonian bulk stress tensors, T1_TRM through T5_TRM, now labeled T1_all through T5_all, are consistent with more fault-slip data than those defined by the TRM, and in some SSs, the explained fault-slip data are now equal to or greater than four (

Table 4. The activated fault-slip data under the bulk enhanced Andersonian stress tensors with MA ≤ 30°.

ST	N	σ1	σ2	σ3	R	ST REG	MMA (MA ≤ 30°)	FT (MA ≤ 30°)	N (Appendix A)	SS
T1all	92	143°–00°	143°–90°	053°–00°	0.18	TRP-SS	16°	18	14, 23, 41, 43, 44, 45, 46, 50, 52, 54, 69, 72, 77, 78,79, 87, 91, 92	SS2 (1), SS3(1), SS6 (6), SS7 (7), SS11 (3)
T2all	92	103°–90°	013°–00°	103°–00°	0.60	PE	13.5°	19	14, 25, 26, 33, 34, 38, 40, 52, 54, 59, 66, 70, 74, 75, 80, 86, 89, 90, 91	SS2 (1), SS3 (4), SS6 (2), SS7 (8), SS11 (4)
T3all	92	175°–90°	085°–00°	175°–00°	1.00	TRN	14°	22	1, 9, 17, 19, 23, 28, 30, 31, 33, 38, 39, 60, 62, 63, 65, 68, 73, 75, 76, 82, 85, 90	SS2 (2), SS3 (7), SS6 (2), SS7 (9), SS11 (2)
T4all	92	063°–00°	063°–90°	153°–00°	0.20	TRP-SS	12.4°	27	3, 4, 5, 7, 8, 9, 10, 12, 13, 15, 16, 18, 36, 38, 42, 57, 58, 60, 62, 63, 64, 65, 70, 82, 83, 84, 85	SS2 (11), SS3 (2), SS6 (2), SS7 (9), SS11 (3)
T5all	92	019°–00°	109°–00°	019°–90°	0.00	TRP	15.5°	17	3, 4, 21, 22, 29, 35, 38, 47, 48, 49, 53, 58, 61, 66, 85, 87, 88	SS2 (2), SS3 (4), SS6 (4), SS7 (4), SS11 (3)

The activated fault-slip data under the resolved bulk enhanced Andersonian stress tensors, namely T1_all through T5_all, considering only the MA≤ 30° criterion used in the best-fit stress inversion methods. TRP: Transpression, SS: (pure) Strike-Slip, TRN: Transtension. PE: Pure Extension. Explanation as in Table 1, Table 2 and Table 3.

). For example, in SS7, the T4_all tensor explains nine instead of four fault-slip data explained by the T4_TRM, and T2_all explains eight instead of six fault-slip data explained by the T2_TRM; and in SS11, the T2_all tensor explains four fault-slip data, although no Andersonian site stress tensor was able to be defined in this SS. In addition, the MMA in each solution is less than 20°. This is a fact that fortifies the fault-slip data homogeneity and that leads to accepted resolved stress tensors.

The fifth analysis stage deals with the fault-slip data explained by the enhanced Andersonian stress tensors, i.e., T1_all through T5_all, since some of the latter define the TRP and TRN stress regimes. Do these fault-slip data define Andersonian or non-Andersonian stress tensors by applying a best-fit stress inversion method (i.e., a method with no limitation of having one principal stress axis in a vertical position)?

The ‘Minimized Shear Stress Variation,’ based on the algorithm of [81,82], defines stress tensors labeled T1_MF to T5_MF (Figure 4,

Table 5. Best-fit stress tensors from the fault-slip data driven by the enhanced Andersonian bulk stress tensors.

ST	n	σ1	σ2	σ3	R	ST REG	MMA (MA ≤ 30°)	FT (MA ≤ 30°)	N (Appendix A)	SS
T1MF	18	144°–03°	014°–86°	234°–03°	0.46	SS	14.6°	16	14, 41, 43, 44, 45, 46, 50, 52, 69, 72, 77, 78,79, 87, 91, 92	SS2 (1), SS6 (6), SS7 (6), SS11 (3)
T2MF	19	321°–88°	196°–01°	106°–02°	0.60	PE	12.6°	19	14, 25, 26, 33, 34, 38, 40, 52, 54, 59, 66, 70, 74, 75, 80, 86, 89, 90, 91	SS2 (1), SS3 (4), SS6 (2), SS7 (8), SS11 (4)
T3MF	22	263°–04°	105°–86°	353°–02°	0.84	SS-TRN	14.2°	21	1, 9, 17, 19, 23, 28, 30, 31, 33, 38, 39, 60, 62, 63, 65, 68, 73, 75, 76, 82, 85, 90	SS2 (2), SS3 (7), SS6 (2), SS7 (9), SS11 (2)
T4MF	27	244°–03°	341°–69°	153°–20°	0.29	TRP-SS	10.4°	27	3, 4, 5, 7, 8, 9, 10, 12, 13, 15, 16, 18, 36, 38, 42, 57, 58, 60, 62, 63, 64, 65, 70, 82, 83, 84, 85	SS2 (11), SS3 (2), SS6 (2), SS7 (9), SS11 (3)
T5MF	17	202°–13°	304°–41°	098°–45°	0.16		15.1°	17	3, 4, 21, 22, 29, 35, 38, 47, 48, 49, 53, 58, 61, 66, 85, 87, 88	SS2 (2), SS3 (4), SS6 (4), SS7 (4), SS11 (3)

The resolved bulk stress tensors after applying the best-fit stress inversion method, “Minimized Shear Stress Variation,” which uses the algorithm of [81,82] onto the T1_all through T5_all fault-slip data. TRP: Transpression, SS: (pure) Strike-Slip, TRN: Transtension. PE: Pure Extension. Explanation as in Table 1, Table 2 and Table 3.

). The resolved best-fit stress tensor, T5_MF, defined by the same fault-slip data with the minimization criterion MA ≤ 30° and almost the same MMA, is non-Andersonian, i.e., having its principal stress axes plunging less than 65° [86].

In the final part of our analysis (sixth stage), we tackle another issue concerning the above-described T1_all through T5_all stress tensors. It concerns the degree of similarity among the resolved stress tensors. This issue has been pointed out by [44,83,84], who suggested the comparison and the degree of similarity between two resolved stress tensors A and B with the use of the Stress Tensor Discriminator Faults (STDF), i.e., the faults which were activated by either A or B stress tensor, but not from both. In particular, three percentages of the Stress Tensor Discriminator Faults are calculated: one for the AB fault-slip dataset, and the other for the A and B fault-slip datasets, respectively. None of the calculated enhanced Andersonian bulk stress tensors are similar, since the percentages of the STDFs are above 80% (

Table 6. Similarity between the enhanced Andersonian bulk stress tensors.

STDF(%)	T2all	T3all	T4all	T5all
T1all	(87.9)(77.8)(78.9)	(97.4)(94.4)(95.5)	(100)(100)(100)	(97.1)(94.1)(94.4)
T2all		(89.2)(81.8)(78.9)	(95.5)(92.6)(89.5)	(94.1)(88.2)(89.5)
T3all			(80.5)(70.4)(63.6)	(94.6)(88.2)(90.9)
T4all				(87.2)(81.5)(70.6)

Comparison between the calculated stress tensors (e.g., A and B) of the present analysis using the TRM [44]. The comparison is carried out using a priori the Stress Tensor Discriminator Faults (STDF) [44,83,84] and an MA ≤ 20°. The bracketed numbers in rows show the percentages of the STDF of the dataset driven by AB, A, and B stress tensors, respectively. A: The stress tensors in rows; B: the stress tensors in columns.

), suggesting that they define distinct stress regimes that should be explained from a geological point of view.

5. Structural Interpretation—Discussion

5.1. Insight into the Paleostress Analysis from Previous Approaches

The paleostress analysis presented by [32] deals with fault-slip data that have been recorded in ultramafic rocks, i.e., rocks easily fractured, forming fault rocks as reported by [32], making the ‘no-fault interaction’ assumption of the stress inversion methods highly uncertain, especially in the limited spatial scale of the SSs. When heterogeneous fault-slip data are recorded in an SS of limited spatial scale, there is always the question of whether the slip independence among the recorded fault-slip data are valid. This is especially true for fault planes interconnected to each other and that are part of a prominent fault structure or a fault zone, a fact that strongly influences the validity of the stress inversion techniques [80].

The resolved best-fit site stress tensors of [32] are characterized by considerable diversity in the orientation of the principal stress axes and the stress ratio (see their Figure 5). Similar results have been found in our first analysis stage by applying the [81,82] best-fit stress inversion method with the aid of MyFault software. However, the very small number of the fault-slip data, i.e., < 6 fault-slip data, from which these tensors were defined, constitutes these solutions as potentially resulting from randomly generated faults [36].

Any resolved site stress tensor calculated mathematically with a best-fit stress inversion software results in a specific combination of fault activations. As a result, it is not only the stress regime, but also the fault activations that should be geologically reasonable and plausible. Are these fault activation combinations similar when determined by different software? As shown in Figure 5 of [32], in the SS3 station (3—N of Palo), the Andersonian and non-Andersonian tensors, T1 and T2, respectively (as derived using FSA), are grouped in the same event B, whereas with Win-Tensor software, the resolved stress tensors have been grouped differently due to different fault combinations.

In a more detailed inspection of the above-mentioned best-fit stress tensors, it is evident that they can be either Andersonian or not, considering if the steepest plunging principal stress axis plunges ≥ 65° or not, respectively (cf. [86]). Nonetheless, in most cases, the non-Andersonian stress tensors have their greatest or least principal stress axes in a (sub)horizontal position. An explanation for this is that the four components of the reduced stress tensor are defined using best-fit algorithms simultaneously but independently from one another, and without having any physical, and therefore, geological constraints or limitations concerning the position of the principal stress axes. However, non-Andersonian stress tensors can be falsely resolved from fault-slip data if the latter are heterogeneous [85] or mixed in kinematics, i.e., dip-slip (thrust or normal), oblique, and strike-slip faults driven by transpressional s.l. stress regimes. It is worth noting that in TRP or TRN, where R ≤ 0.125 and R ≥ 0.875, respectively [79], it is not important which of the two principal stress axes is in a vertical position, since their magnitudes are very close, if not equal, making it trivial as to whether the resolved stress type is Andersonian or not.

Another issue is that the variations in the orientations of the stresses and stress ratios of the best-fit site stress tensors imply strong stress perturbations for the region. Unfortunately, in such cases, stress perturbations cannot be established based solely on the best-fit stress inversion methods at the different SSs in the case of heterogeneous fault-slip data [85].

The iterative process followed in the FSA and Win-Tensor software in each SS results in calculating, solely and strictly, the first stress tensor from the original fault-slip data. In contrast, the next resolved stress tensors are biased to different degrees, since they were calculated from successively nested datasets, i.e., smaller in number than the original dataset, especially if the SS is of limited spatial scale. Moreover, this inconsistency cannot be seen by the best-fit stress resolved site stress tensors themselves if the latter are diverse in terms of the orientation of the stresses and the stress ratios due to the different input fault-slip data combinations recorded at the different SSs.

5.2. Insight into the TRM Site and Bulk Enhanced Andersonian Stress Tensors

The TRM application calculates enhanced Andersonian site stress tensors in all SSs, except for SS11. Most define TRP and TRN stress regimes, as implied by the predominance of the strike-slip and oblique-slip faults in the study area. In cases where the site stress tensors were calculated from less than 10 fault-slip data, they present differences in the horizontal stress axis orientation like the pairs (T1_SS3, T_SS6) and (T3_SS3, T2_SS7) (Table 2).

The TRM application on the whole fault-slip dataset (Appendix A) defines five bulk stress tensors, T1_TRM, through T5_TRM (Table 3). Apart from the first, the other bulk stress tensors were calculated with more than 10 fault-slip data, i.e., the minimum number to overcome randomly defined stress tensors [41]. These bulk stress tensors can activate more than four differently oriented fault-slip data in a few SS, and fault-slip data from several SS having similar attitudes, as in the case of similar fault activations recorded through focal mechanisms at different sites in seismically active areas, e.g., [87]. In addition, from Table 2 and Table 3, it seems that both TRM site and bulk stress tensors define similar stress regimes like (a) T1_TRM with T_SS6, (b) T2_TRM with T3_SS7, (c) T3_TRM with T3_SS3 and T2_SS7, (d) T4_TRM with T_SS2 and T1_SS7, and (e) T5_TRM with T1SS3. This similarity occurs because the TRM stress tensors are calculated from the Tensor Ratio Compatible Faults regardless of the number of fault-slip data [44,85].

The interesting issue is that all the resolved enhanced Andersonian site stress tensors define transpressional s.l. stress regimes, i.e., those that fit well with the predominance of subhorizontal to oblique-slip faults over the dip-slip faults [24,25,26] in the recording dataset mentioned. Therefore, the enhanced Andersonian stress tensors can fairly describe transpression s.l. stress regimes.

When the minimization criterion of MA ≤ 30° was only considered, the TRM bulk stress tensors explained more fault-slip data out of the whole fault-slip dataset than those defined by the TRM application (see T1_all through T5_all in Table 4). Because of the large number of the explained fault-slip data, these solutions can hardly be considered to be a result of combining random fault-slip data by taking into account only the MA minimization criterion used by the best-fit stress inversion methods. For example, the T1_all stress tensor explains 10 more fault-slip data than T1_TRM. In addition, when a best-fit stress inversion method was applied to the fault-slip data explained by the T1_all to T5_all tensors, the resolved bulk stress tensors T1_MF through T5_MF, apart from the T5_MF, did not differ significantly from the T1_all through the T5_all, respectively (Table 4 and Table 5). Apart from the T5_MF, all the tensors define one principal stress axis at ≤ 25° from the vertical position following Anderson’s assumed ‘standard state’ stress configuration near the Earth’s free surface [88]. On the other hand, considering the T5_MF, someone might promptly conclude that the driving stress regime in the region might be of non-Andersonian type. However, the Andersonian standard state of stress is also supported by the paleomagnetic data in the region and, more precisely, the inclination values calculated by [69], which in their majority show no significant rotations (<10°) around the horizontal axes. As a result, both the Andersonian (T5_all) and the non-Andersonian (T5_MF) stress tensors explain the same fault-slip data with the minimization criterion MA ≤ 30° and almost the same MMA. However, the enhanced Andersonian tensor T5_all is more admissible for the region, suggesting that the enhanced Andersonian stress tensors can fairly describe the faulting deformation in a region, even if the latter is subjected to transpression s.l. tectonics.

5.3. Remarks on Comparing the Methods

The detailed re-examination of these fault-slip data with the TRM application, which uses additional constraints induced using Slip Preference Analysis, likewise defines TRM site stress tensors with less than 10 fault-slip data [32] and TRM bulk stress tensors with more than 10 fault-slip data (except for T1_TRM, which was calculated from eight fault-slip data). However, the TRM site and bulk stress tensors are quite similar and not diverse, in terms of stress axis orientation and stress ratio, as those calculated from the best-fit stress inversion methods. It also indicates that the TRM defines stress tensors that are not as sensitive to the input fault-slip data as the best-fit stress tensors are. Thus, we can argue that the best-fit stress inversion methods suffer from the ‘overfitting’ modelling error, which is more limited in the TRM application due to the additional SPA geological constraints [24,25,26].

More importantly, the resolved bulk stress tensors obtained from the TRM were determined exactly from the same original fault-slip dataset. These TRM bulk stress tensors were defined with much more than 10 fault-slip data, indicating that these solutions can hardly be considered a result of random fault-slip data (see Table 4) [41]. TRM bulk stress tensors also explain more fault-slip data recorded in more SSs. As a result, the basic assumption of the inverse problem, i.e., that the faults slip independently, does not seem to be violated when the ‘point’ no longer occupies an infinitesimal size, as defined in physics, or the SS of limited spatial scale, as defined in geology, but refers to a geological region covering several tens of km². In the case of bulk stress tensors, the large number of fault-slip data recorded in different SSs, as well as the fact that the dataset, as a whole, represents a wide variation, both in the orientation and size of the measured faults, eliminate, as much as possible, any influence imposed by the fault structures themselves. Likewise, it eliminates any deviation between the local and regional kinematic field, e.g., in a thrust belt [89].

6. A New Paleostress History and Tectonic Implications

The type of deformation and paleostress history, and therefore, the late-orogenic deformation in the area north of Corsica Island, is poorly constrained and still under debate because it is an area where both the Pyrenean and western Alpine domains have come into contact since the Late Eocene [90]. Large-scale kinematics are dominated by the coeval effects of the ending collision of Iberia with Eurasia, the Apulia (Adria)-Eurasia convergence, and the north dipping subduction of Africa along the southern margin of the Iberian plate [45]. The overlap of several tectonic events [57,66], including the Early–Middle Miocene oceanic spreading of the Liguro-Provençal basin, the coeval 50° counterclockwise rotation [69] of the Ligurian Alps-TPB system [91], and the drifting of the Corsica-Sardinia block [92] make the area very complicated. Likewise, in the Ligurian Alps, there are two different scenarios, which, however, are based on studies having analyzed different structural data, thus making their comparison hard. The first is of [93], who suggested an E-W extensional regime with lithospheric thinning that was active since the Early Oligocene, and the second of ([58], and references therein), who suggested a transpressive regime for the tectonic evolution of the Ligurian Alps since the Oligocene.

Our paleostress analysis results are outlined in the following stress regimes:

T1_TRM is a Transpression-(pure) Strike-Slip (TRP-SS) stress regime with a NW-SE trending σ₁ axis in a similar orientation to the Rupelian−Early Chattian event A of [32] and the Late Eocene–Early Oligocene compressional D3 event of [73]. On the other hand, [66] argued that the NW-SE compression is much younger, starting from the Late Miocene (post-Tortonian), as powered by the displacement of the Adriatic indenter against Eurasia and following the NE-SW-trending prevailing compressional regime that affected the Tertiary Piedmont Basin evolution in the Late Oligocene–Early Miocene. T1_TRM stress tensor activated mainly ENE-WSW striking right-lateral, oblique-to-strike-slip contractional faults, but not reverse faults like those mentioned by [73]. If this is not due to fault sampling, these faults indicate that the ENE-WSW striking detachment shown in Figure 1 might have activated as a strike-slip fault under the T1_TRM stress tensor. ENE-WSW trending faults, similar to those activated under the T1_TRM stress tensor, limit the Tertiary Piedmont Valley in the area of Pra’ Vallarino and north of Palo, as well as in the north-eastern boundary of the Voltri Massif [57], and they have been traced using photo-lineaments [74]. The activation of strike-slip instead of reverse faults in the region might indicate spatial variations of the driving stresses among the different regions.

T2_TRM is an E-W Pure Extension (PE) stress regime that cannot be correlated with [32]‘s proposed events. In other words, this extension has not been defined by [32]‘s analysis that used both the FSA and Win-Tensor software. However, it fits well with the first brittle deformation phase (D4) of [57], which was a regional E-W to NE-SW extension developed in the metamorphic basement and the Tertiary Piedmont Basin in Rupelian–Early Chattian (~34–26 Ma), and led to the opening of the Liguro-Provencal Ocean, along with a counterclockwise rotation and drifting of Corsica-Sardinia [48]. The exhumation of the Voltri Massif to greenschist-facies, which took place approximately 34-30 Ma [94], has been attributed to E-W extension by [60]. The latter authors consider the E-W extension as an orogen parallel extension to the Southern Alps due to the advancing of the Western Alps to the west and the retreat of the Apennines slab. Moreover, [90] suggested such an extension as a post-orogenic one that truncated the Pyrenean thrusting and formed the Liguro-Provencal and Tyrrhenian basins [95], showing a major change in subduction dynamics [51].

T3_TRM defines a TRN stress regime that cannot correlate with [32]‘s events. On the other hand, it can be correlated with some N-S tensional stresses detected in several parts of the Tertiary Piedmont Basin, especially in its SE portion, though its timing relation with the other stress regimes is unclear [66]. These N-S tensional stresses are compatible with relatively small ENE-WSW to WNW-ESE trending normal faults mapped in the SE parts of the Tertiary Piedmont Basin [65,96,97], similar in strike to those activated by the T3_TRM stress tensor (Figure 2d). [57] also mentioned a transtensional stress regime in the Late Chattian representing the transition from Early Oligocene rifting-related extension to the Early Miocene rotation-related transpression. In particular, [55] stated that this transtension marks the time when the Apennines start to ‘pull’ the southern Western Alps northeastward, producing a large shear zone (i.e., the Ligurian Alps-TPB) between the two chains. In this setting, the Tertiary Piedmont Basin acted as a strongly subsiding piggyback basin above rotating thrust sheets associated with the regional rotation caused by the oceanic spreading of the Liguro-Provençal basin, the Corsica-Sardinia drifting, and the eastward retreat of the northern Apennines slab.

T4_TRM (ENE-WSW, TRP) and T5_TRM (NNE-ESE, TRP) can be correlated with event B of [32], dated in Early Miocene (Aquitanian-early Burdigalian (?)) by them in the Late Oligocene–Early Miocene in the Tertiary Piedmont Basin by [66], and in Aquitanian–Serravallian (~23–12 Ma) by [57]. It is worth noticing that both T4_TRM and T5_TRM stress tensors reveal transpressive deformation. In addition, T4_TRM and T5_TRM stress tensors, with the former preceding the latter, fit well with the ~40° counterclockwise rotation of the σ₁ axes from ENE-WSW to NNE-SSW trends [73]. Similarly, the T4_TRM and T5_TRM stress tensors agree with the paleomagnetic data results that reveal ~50° of counterclockwise rotation for the Tertiary Piedmont Basin and the underlying Ligurian basement in the Aquitanian–Serravallian times [69]. Field studies and seismic profile interpretations in the area provide strong evidence for transpressional structures and strike-slip flower structure geometries along E-W to ENE-striking faults that truncate and offset several unconformities bounding the Oligocene to lower Miocene sedimentary sequence, which in turn are sealed by Upper Miocene sediments [58,98].

The five (5) bulk stress tensors, T1_TRM through T5_TRM (Table 3), are well distinguished from each other and can be well correlated with the TRM site stress tensors (Table 2). Most TRM bulk stress tensors define TRP and TRN stress regimes instead of a (pure) strike-slip regime (SS). These stress regimes suggest that the deformation in the study area, which was part of the boundary between the orogenic belts of the Alps and Apennines, was not controlled by any large-scale (pure) strike-slip fault zone like the Sestri-Voltaggio Zone (Figure 1b), and was distributed along numerous and smaller strike-slip faults. Therefore, we consider the area to be a well-faulted relay ramp at a contractional overstep between larger-scale right-lateral strike-slip faults [99].

We have to note that estimated stress tensors of [32] are determined in some SSs from < 10 fault-slip data, and without any prior restoration or rotation of the recorded fault-slip data, although a ca. 50° of counterclockwise rotation for the Tertiary Piedmont Basin and the underlying Ligurian basement has been suggested in the Aquitanian–Serravallian times [69]. Since no absolute age information is available, as [32] pointed out, we compare the T1_TRM through T5_TRM stress tensors with the suggested deformational events of [32] and other published information concerning the deformation of the region, as described above in the five stress regime outlines.

Conclusively, the TRM application on the fault-slip data of [32] defines (a) a TRP-SS stress regime (T1) with NW-SE contraction, which can be correlated with the Late Eocene deformation of [73]; (b) an Oligocene E-W extensional regime (T2) fitting well with the E-W extension mentioned for the broader area due to a major change in subduction dynamics, perhaps as a consequence of collision and slowing down of the northward motion of Africa [51,89]; (c) an N-S transtension (T3), which has the σ₃ axis perpendicular to the σ₃ axis of the T2 stress regime; and (d) Miocene transpression (T4) with ENE-WNW contraction that changes to (e) transpression with NNE-SSW contraction (T5).

Based on these, we can argue that the Late Oligocene–Early Miocene marks a crucial period for the stress evolution in the broader area, since the plate convergence transferred eastwards from the Pyrenees collision zone to the Corsica-Sardinia trench system and the stress regime in the Ligurian Alps shifted gradually from an NW-SE to NE-SW direction of compression [57,90,92].

In particular, considering that the 50° counterclowise rotation occurred from the T4 to T5 stress regime, the T1 through T3 stress regimes should be restored to pre-rotational orientations. In that sense, the restored T1 represents a TRP-SS stress regime with contraction trending NNE-SSW. Such a contraction fits well with the NW-SE contraction in the Western Alps and its change to N-S towards the Pyrenees, placing the region along the Western Alps and Pyrenees (Figure 5a). The restored T2 stress regime is an extension trending NW-SE, similar to the extension dominating the Liguro-Provencal rifting (Figure 5b). The restored T3 stress regime, which is a transtension having a NE-SW trending σ₃ axis, i.e., perpendicular to the NW-SE T2 σ₃ axis, might be considered a local mutual permutation of the T2 stresses. The T4 and T5 stress regimes shown in Figure 5c,d indicate the gradual rotation that took place from the Early until the Middle Miocene. Since then, the area’s configuration has been characterized by a gradual counterclockwise rotation and complex oblique and strike-slip deformation, as is documented in the upper crust via seismic tomography and earthquake alignments [100].

The change from the Oligocene extension to the Late Oligocene-Miocene compression and, in general, the stress reorganization in the Ligurian Alps, i.e., the internal area of the Northern Apennines and the Western Alps, can be attributed to the decrease in the retreating rate of the Apennines slab.

7. Conclusions

The present analysis shows that applying best-fit stress inversion methods to heterogeneous fault-slip data, even when complemented with sophisticated tools such as the Monte Carlo search method, does not ensure that the resolved stress tensors would represent the geologically real driving stress tensors. This conclusion is more profound when these methods are implemented on the fault-slip data of a limited number and at stations of limited spatial scale. The resolved best-fit site stress tensors are characterized by high diversity in stress orientation and at different stress ratios, with the tendency to define non-Andersonian (oblique) stress axes, especially when this approach is performed on a small number of recorded fault-slip data that might constitute a dynamically mixed fault-slip dataset. As a result, it is very hard and ambiguous to gather and to interpret fault-slip data under the same regional stress regime by considering only the site stress tensor solutions. On the contrary, the TRM when using additional fault-slip data separation criteria seems more promising when dealing with heterogeneous fault-slip data in finding enhanced Andersonian bulk stress tensors that describe the paleostress history and fault activation under the different faulting events, even in the absence of tectonostratigraphic criteria.

The TRM bulk paleostress tensors provide a more constrained paleostress history for the Voltri Massif and the Ligurian Alps, comprising five distinct stress regimes, which, after the restoration of the ~50° CCW rotation, comprise: (a) a transpression–strike-slip stress regime (T1) with NNE-SSW contraction in the Late Eocene, (b) an Oligocene NW-SE extensional regime (T2), which fits with the NW-SE extension documented for the broader area north of Corsica due to a significant change in subduction dynamics, (c) a transient, local, or ephemeral NE-SW transtension (T3), which might be considered to be a local mutual permutation of the T2 stresses, and (d) a Miocene transpression with a contraction that progressively shifted from ENE-WSW (T4) to NNE-SSW (T5). This paleostress history reflects the stress reorganization in the Ligurian Alps due to the combined effect of the transfer of the plate convergence eastwards, from the Pyrenees to the Corsica-Sardinia trench system, and the decrease in the retreating rate of the northern Apennines slab.