Heterogeneous Fault-slip Data Research Articles

Paleostress reconstruction of the Chaling-Yongxing basin in the central South China Block: Bulk (areal or multisite) vs. site stress tensors

A critical problem in fault-slip data stress analysis is separating the heterogeneous fault-slip data into homogeneous subsets. Recent attempts to solve this problem use tectonostratigraphic criteria based on the age of the sites where site stress tensors are calculated with a best-fit stress inversion method. The present study calculates bulk (areal or multisite) stress tensors from fault-slip data published in the Chaling-Yongxing Basin (CYB) in central South China with the Tensor Ratio Method (TRM). These tensors are compared with published site stress tensors to examine whether bulk or site stress tensors define regional or local stress regimes and similar or not paleostress history. Two regional stress regimes for the Late Cretaceous-Paleogene period, describing an older Late Cretaceous stage with transpression-strike-slip (TRP-SS) tectonics (NNE-SSW contraction and WNW-ESE extension) and a younger Paleogene stage with radial-pure (RE-PE) extensional tectonics (WNW-ESE extension), are defined. These stages are interpreted to be transitive through a transtension (TRN) tectonics of similar WNW-ESE extension and responsible for the formation and evolution of the CYB under tectonic processes related to the India-Asia collision and the retreat of the (Paleo)Pacific plate. The present analysis shows that tectonostratigraphic criteria based on the age of the exposed rocks might be insufficient for the coherent correlation of stress tensors resolved in sites of limited size and, thus, for unraveling the paleostress history of a region.

A neural network approach for classification of fault-slip data in geoscience

In geoscience, paleostress studies are a vital tool for understanding the tectonic evolution of the region. The collected hundreds or even thousands of heterogeneous fault-slip data need to be divided into homogeneous (i.e., belonging to similar tectonic environments) subgroups by geologists. Computer-based paleostress inversion programs are able to run homogenous data sets. Here, we are aiming for the classification of heterogeneous fault-slip data into homogeneous sub-data which is performed by using several techniques in Machine Learning (ML) algorithms, which are Artificial Neural Network (ANN), Naïve Bayes (NB) and Logistic Regression (LR) models. When these models are executed on the Anaconda Navigator interface with python language, the accuracies are obtained as 87.17 % for ANN, 79.71 % LR, and 62.1 % NB.

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)

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.

Tectonic evolution of the northern Verkhoyansk Fold-and-Thrust Belt: insights from palaeostress analysis and U–Pb calcite dating

AbstractA combined structural and geochronological study was carried out to identify the tectonic evolution of the northern Verkhoyansk Fold-and-Thrust Belt, formed on the east margin of the Siberian Craton during late Mesozoic collision. Fault and fold geometries and kinematics were used for palaeostress reconstruction along the Danil and Neleger rivers cross-cutting the central and western parts of the Kharaulakh segment of the northern Verkhoyansk. Three different stress fields (thrust, normal and strike-slip faulting) were identified after separation from heterogeneous fault-slip data. Thrust and normal faulting stress fields were found in both areas, whereas a strike-slip faulting stress field was only found in Neoproterozoic rocks of the Neleger River area. U–Pb laser ablation – inductively coupled plasma – mass spectrometry (LA-ICP-MS) dating of calcite slickenside samples reveals the following succession of major deformation events across the northern Verkhoyansk: (i) The oldest tectonic event corresponding to the strike-slip faulting stress field with NE–SW-trending compression axis is Early Permian (Cisuralian, 284 ± 7 Ma) and likely represents a far-field response to the late Palaeozoic collision of the Kara terrane with the northern margin of the Siberian Craton. (ii) A slickenfibrous calcite age of 125 ± 4 Ma is attributed to the Early Cretaceous compression event, when the fold-and-thrust structure was formed. (iii) U–Pb slickenfibre calcite ages of 76–60 Ma estimate the age of a Late Cretaceous – Palaeocene compression event, when thrusts were reactivated. Slickensides related to both (ii) and (iii) compressional tectonic events formed by similar stress fields with W–E-trending compression axes. (iv) From the Palaeocene onwards, extensional tectonics with approximately W–E extension predominated.

HGA: A genetic algorithm method for direct estimation of paleostress states from heterogeneous fault-slip data

Field data on fault-slip observations is commonly heterogeneous. Paleostress estimation from such data sets is, in general, carried out in two steps: (i) the classification of the heterogeneous data set into homogeneous subsets and (ii) an inversion of each homogeneous subset. This study gives a new approach, the HGA, that combines the two issues in a single step process and gives the stress tensors directly. The given heterogeneous data are directly operated upon by the genetic algorithm operators, initialization, elitism, selection, encoding, crossover and mutation. These operations simulate such a guided search that finds successively fitter solutions, the stress tensors, until the globally fittest solution is obtained. We first explain the basic steps of the algorithm on a working example and then demonstrate its veracity using several synthetic and two natural examples.The proposed genetic algorithm method obviates the necessity of having first to classify the heterogeneous data into homogeneous sets. It directly estimates different stress states by inversion of the given heterogeneous fault-slip data. In contrast to the existing linear methods, the method is not vulnerable to entrapment of the solution in a local optimum. Although the method requires an a priori estimate of the maximum number of expected homogeneous sets in a given population, this estimate does not control the final results. Like any other method, the genetic algorithm method too has its merits and limitations and these are discussed.

Is the Monte Carlo search method efficient for a paleostress analysis of natural heterogeneous fault-slip data? An example from the Kraishte area, SW Bulgaria

The efficiency of the Monte Carlo search method (MC) in separating successfully heterogeneous fault-slip data is evaluated using the separation and stress inversion TR method (TRM). Fault-slip data from the Kraishte area of SW Bulgaria have been used from which two extensional stress regimes with stress axes oriented as in Andersonian state of stresses have been published with the MC. The examination with the TRM that is based on constraints from the Slip Preference Analysis (SPA), indicates that the MC in the absence of any tectonostratigraphic criteria, might well constrain the orientation of the stress axes, but fails to correctly separate the fault-slip data into homogeneous groups, and to correctly define the stress ratio of the stress tensors. The problem is enlarged when a small number of fault-slip data is used for the determination of the stress tensor as occurs in sites of limited size. The resolved stress tensors vary significantly from site to site, because they strongly depend on the number of the recorded fault-slip data and their combinations and, more precisely, on the number of the non-TR compatible faults (i.e. faults without geometric, kinematic and dynamic compatibility according to SPA) that have been considered in their calculation.

The use of Stress Tensor Discriminator Faults in separating heterogeneous fault-slip data with best-fit stress inversion methods. II. Compressional stress regimes

Synthetic heterogeneous fault-slip data as driven by Andersonian compressional stress tensors were used to examine the efficiency of best-fit stress inversion methods in separating them. Heterogeneous fault-slip data are separated only if (a) they have been driven by stress tensors defining ‘hybrid’ compression (R < 0.375), and their σ1 axes differ in trend more than 30° (R = 0) or 50° (R = 0.25). Separation is not feasible if they have been driven by (b) ‘real’ (R ≥ 0.375) and ‘hybrid’ compressional tensors having their σ1 axes in similar trend, or (c) ‘real’ compressional tensors. In case (a), the Stress Tensor Discriminator Faults (STDF) exist in more than 50% of the activated fault slip data while in cases (b) and (c), they exist in percentages of much less than 50% or not at all. They constitute a necessary discriminatory tool for the establishment and comparison of two compressional stress tensors determined by a best-fit stress inversion method. The best-fit stress inversion methods are not able to determine more than one ‘real’ compressional stress tensor, as far as the thrust stacking in an orogeny is concerned. They can only possibly discern stress differences in the late-orogenic faulting processes, but not between the main- and late-orogenic stages.

The use of Stress Tensor Discriminator Faults in separating heterogeneous fault-slip data with best-fit stress inversion methods

A widely accepted best-fit stress inversion method is applied on synthetic extensional heterogeneous fault-slip data generated by two driving Andersonian stress tensors having similar stress ratios, and σ3 axes with different trends, in order to examine whether the misfit angle (MA) minimization criterion can be used for separating heterogeneous fault-slip data. The examination shows that the resolved best-fit stress tensors have σ3 axes that tend to the bisector of the σ3 axes of the driving stress tensors, and stress ratios smaller than those of the latter. Moreover, there is a tendency towards determining radial extension stress regimes, although such regimes are rare in the Earth's crust. More importantly, the best-fit stress inversion methods that use solely the MA minimization criterion cannot be used for the separation of heterogeneous fault-slip data, especially when the extensional driving stress tensors have stress ratios smaller than R = 0.5, i.e., as are the favored paleostresses in the Earth's crust. In contrast, the percentage of the Stress Tensor Discriminator Faults (STDF) can be a very useful discriminator tool for the establishment and comparison between two resolved stress tensors as the latter have been determined by a best-fit stress inversion method. Moreover, the existence of fault-slip data that can been considered as possible STDFs during the recording stage advocate for the heterogeneous origin of the fault-slip dataset.

Fault-slip inversions: Their importance in terms of strain, heterogeneity, and kinematics of brittle deformation

Heterogeneous deformation is intrinsic in natural deformation, but often underestimated in the analysis and interpretation of mesoscopic brittle shear faults. Based on the analysis of 11,222 faults from two distinct tectonic settings, the Central Andes in Argentina and the Sudbury area in Canada, interpolation of principal strain directions and scaled analogue modelling, we revisit controversial issues of fault-slip inversions, collectively adhering to heterogeneous deformation. These issues include the significance of inversion solutions in terms of (1) strain or paleo-stress; (2) displacement, notably plate convergence; (3) local versus far-field deformation; (4) strain perturbations and (5) spacing between stations of fault-slip data acquisition. Furthermore, we highlight the value of inversions for identifying the kinematics of master fault zones in the absence of displaced geological markers. A key result of our assessment is that fault-slip inversions relate to local strain, not paleo-stress, and thus can aid in inferring, the kinematics of master faults. Moreover, strain perturbations caused by mechanical anomalies of the deforming upper crust significantly influence local principal strain directions. Thus, differently oriented principal strain axes inferred from fault-slip inversions in a given region may not point to regional deformation caused by successive and distinct deformation regimes. This outcome calls into question the common practice of separating heterogeneous fault-slip data sets into apparently homogeneous subsets. Finally, the fact that displacement vectors and principal strains are rarely co-linear defies the use of brittle fault data as proxy for estimating directions of plate-scale motions.

Stress field reconstruction in an active mudslide

Meso-scale structures from gravitational slope deformation observed in landslides and deep-seated gravitational slope failures are very similar to those of endogenous ones. Therefore we applied palaeostress analysis of fault-slip data for reconstructing the stress field of an active mudslide in Pechgraben, Austria. This complex compound landslide has developed in clayey colluvium and shale and was activated after a certain period of dormancy in June 2013. During the active motion on June 12, 2013, 73 fault-slip traces at 9 locations were measured within the landslide body. The heterogeneous fault-slip data were processed in term of palaeostresses, the reconstructed palaeostress tensor being characterized by the orientations of the three principal stress axes and the stress ratio (which provides the shape of the stress ellipsoid). The results of the palaeostress analysis were compared to airborne laser scan digital terrain models that revealed dynamics and superficial displacements of the moving mass prior and after our survey. The results were generally in good agreement with the observed landslide displacement pattern and with the anticipated stress regime according to Mohr-Coulomb failure criteria and Anderson's theory. The compressional regime was mostly registered at the toe in areas, where a compressional stress field is expected during previous mass-movement stages, or at margins loaded by subsequent landslide bodies from above. On the other hand, extension regimes were identified at the head scarps of secondary slides, subsequently on bulged ridges at the toe and in the zone of horst-and-graben structures in the lower central part of the main landslide body, where the basal slip surface probably had locally convex character. Strike-slip regimes, as well as oblique normal or oblique reverse regimes were observed at the lateral margins of the landslide bodies. The directions of principal stresses could be used as markers of landslide movement directions. Although the multiphase strain indicators were, in general, relatively poorly preserved, the palaeostress analysis revealed several different deformation phases at some landslide parts related to different evolution episodes of the complex landslide. Palaeostress analysis based on the multiple inversion method is an affordable approach for reconstructing stress fields within landslides in soft material. This method is an inexpensive and objective tool for stress field reconstruction even in heterogeneous stress field conditions where the distinction of different faults generations may be impossible.

TR method (TRM): A separation and stress inversion method for heterogeneous fault-slip data driven by Andersonian extensional and compressional stress regimes

The recently proposed TR method (TRM) which uses the slip preference of the faults to separate heterogeneous fault-slip data in extensional and compressional Andersonian stress regimes, is enhanced so as to determine stress tensors with the use of the Wallace–Bott slip criterion. Published natural fault-slip data from the extensional region of Tympaki, Crete, Greece and artificial fault-slip data modeled from the Chelungpu thrust, activated during the 1999 Chi–Chi earthquake in Taiwan, have been used as case studies. In the first case, the fault-slip data previously considered as homogeneous might actually be of heterogeneous origin as they determine two distinct stress tensors that both fit well with the neotectonic faulting deformation of the region. In the second case, where the fault-slip data belonging to three different subsets are of low diversity, the TRM succeeds in defining the driving stress tensors. The Misfit Angle minimization criterion can adequately separate the fault-slip data between two subsets when the percentage of the “Stress Tensor Discriminator Faults” is higher than approximately 70%.

Fault kinematics and stress fields in the Rwenzori Mountains, Uganda

The Rwenzori Mountains in western Uganda form an active rift-transfer zone in the western branch of the East African Rift System. Here we quantify local stress fields in high resolution from field observations of fault structures to shed light on the complex, polyphase tectonics expected in transfer zones. We apply the multiple inverse method, which is optimized for heterogeneous fault-slip data, to the northern and central Rwenzori Mountains. Observations from the northern Rwenzori Mountains show larger heterogeneity than data from the central Rwenzori, including unexpected compressional features; thus the local stress field indicates polyphase transpressional tectonics. We suggest that transpression here is linked to rotational and translational movements of the neighboring Victoria block relative to the Rwenzori block that includes strong overprinting relationships. Stress inversions of data from the central Rwenzori Mountains indicate two distinct local stress fields. These results suggest that the Rwenzori block consists of smaller blocks.

Comparison of Strain Ellipsoid Shape in the South of Ardabil Range (NW), Based on the Results of the Magnetic Susceptibility Anisotropy and Paleostress Methods

In recent years, the method of magnetic survey as one of the new techniques in geological and geophysical studies is known. In this study to determine the shape of the stress field of the two methods, Anisotropy of Magnetic Susceptibility (AMS) and paleostress have been used. Paleomagnetism is the characteristics of magnetic rocks. Some issues in associated with the past places of continental and oceanic plates can be solved. AMS is one of the paleomagnetism methods that pay to measurement of parameters (which are reflector of the magnetic fabrics rocks). It is presenting an ellipsoid with three-axis perpendicular to each other that defines magnetic ellipsoid. In this regard, the number of 12 stations in different rocks (Jurassic to Quaternary) in the southern region of Ardebil sampling was conducted. In this connection, the study of magnetic fabrics has shown an elliptical magnetic susceptibility with the prolate shape. For the separation of paleostress phases in the Khalkhal area using the analysis of the paleostress based on the study of heterogeneous fault-slip data and sliding lineaments. Firstly, data were picked from 10 stations, and after their analysis, the elliptical shape (prolate) has been determinated. The shape of the ellipsoid, based on AMS and paleostress methods and their results show that in both methods the shape of the stress field is prolate.

Stress inversion of heterogeneous fault-slip data with unknown slip sense: An objective function algorithm contouring method

We propose a new method for stress inversion and separation of principal stress states from heterogeneous fault-slip data. The method is semi-automatic, and is based on the moment method of stress inversion (Fry 1999) in combination with the objective function algorithm (OFA) for stress separation (Shan et al 2003). In the presented routine we randomly partition the heterogeneous fault-slip dataset into subsets ranging between one and six. The number of subsets K represents the number of possible mixed stress states in the fault-slip dataset. For each partition number K, we run the OFA 1000 times. Following this we plot and contour the principal stress axes, corresponding to the minimum value of the objective function for each run, in a stereonet. By evaluating how solution clusters of principal stress axes change with increasing number of subsets K, we are able to determine the number of mixed stress states and their optimal solutions for heterogeneous fault-slip datasets. While the numbers of subsets are underestimated, solution-clusters of principal stress axes represent average stress states. However, once the correct number of subsets is reached, solution clusters align with the slip-generating principal stress axes. The solution clusters then become stable, and overestimating the number of subsets does not significantly alter their orientation. The partition number K when stability is obtained thus determines the number of mixed stress states in the heterogeneous dataset, while the corresponding highest density solution clusters give the best estimate of the slip-generating principal stress axes and corresponding stress shape ratios. The inversion routine is tested and confirmed using synthetic data and fault-slip data from the Gullkista fault in Northern Norway. Because the stress calculation is based on the moment method, the inversion routine is insensitive to the correct assessment of slip sense, and only requires the slip vector and orientation of the fault plane as input. It is therefore a robust method to evaluate the number of mixed stress states and their respective stress tensors for complex heterogeneous fault-slip data.

TR method: a practical tool to analyze focal mechanisms and identify the ‘real’ seismogenic fault of an extensional or compressional shallow earthquake sequence

The recently proposed TR method that separates heterogeneous fault-slip data in extensional and compressional stress regimes is applied on focal mechanisms of a number of earthquake sequences aiming to identify the main seismogenic fault plane. The method only requires the earthquake sequence to be classified as related to an extensional or compressional stress regime. In particular, the nodal planes of the compressional or extensional focal mechanisms have been subdivided into two opposite dipping groups taking into account the average trend of their horizontal P or T axis. In the TR diagrams where we plot the slip deviation (from the dip-slip activation) of the nodal plane versus the angle between the dip direction of the nodal plane and the average trend of the P or T axis, the comparison of the geometric, kinematic, dynamic and mechanical compatibility of the two opposite nodal plane groups is possible. Moreover, a quality number (TRn) is devised in order to assess the two opposite facing groups quantitatively. Five earthquake sequences related with an extensional stress regime, and two related with a compressional stress regime, are processed in order to demonstrate and validate the potential and efficiency of the method. These cases indicate that the TR method can be fairly used as a practical tool to assess which nodal planes are more likely to identify the main seismogenic fault plane. The main advantages of the method are (a) the assessment of the nodal plane groups without the necessity to determine all the parameters of the driving stress tensor, i.e., stress axes and stress ratio, (b) the consideration of the slip tendency.

The TR method: the use of slip preference to separate heterogeneous fault-slip data in compressional stress regimes. The surface rupture of the 1999 Chi-Chi Taiwan earthquake as a case study

Abstract Synthetic contractional fault-slip data have been considered in order to examine the validity of widely applied criteria such as the slip preference, slip tendency, kinematic (P and T) axes, transport orientation and strain compatibility in different Andersonian compressional stress regimes. Radial compression (RC), radial–pure compression (RC–PC), pure compression (PC), pure compression–transpression (PC–TRP), and transpression (TRP) are examined with the aid of the Win-Tensor stress inversion software. Furthermore, the validity of the recently proposed graphical TR method, which uses the concept of slip preference for the separation of heterogeneous fault-slip data, is also examined for compressional stress regimes. In these regimes only contractional faults can be activated, and their slip preferences imply the distinction between “real”, i.e., RC, RC–PC and PC, and “hybrid”, i.e., PC–TRP and TRP stress regimes. For slip tendency values larger than 0.6, the activated faults dip at angles from 10° to 50°, but in the “hybrid” regimes faults can dip with even higher angles. The application of the TR method is here refined by introducing two controlling parameters, the coefficient of determination (R2) of the Final Tensor Ratio Line (FTRL) and the “normal” or “inverse” distribution of the faults plotted within the Final Tensor Ratio Belt (FTRB). The application of the TR method on fault-slip data of the 1999 Chi-Chi earthquake, Taiwan, allowed the meaningful separation of complex heterogeneous contractional fault-slip data into homogeneous groups. In turn, this allowed the identification of different compressional stress regimes and the determination of local stress perturbations of the regional or far-stress field generated by the 1999 Chi-Chi earthquake. This includes clear examples of “stress permutation” and “stress partitioning” caused by pre-existing fault structures, such as the N–S trending Chelungpu thrust and the NE–SW trending Shihkang–Shangchi fault zone.

Slip preference on pre-existing faults: a guide tool for the separation of heterogeneous fault-slip data in extensional stress regimes

Synthetic fault-slip data have been considered in the present paper, in order to examine through a simple graphical manner the validity and use of the widely mentioned and applied criteria such as the slip preference, slip tendency, kinematic (P and T) axes, transport orientation and strain compatibility. The examination and description concern extensional stress regimes whose greatest principal stress axis (σ1) always remains in vertical position as in Andersonian stress states. In particular, radial extension (RE), radial–pure extension (RE–PE), pure extension (PE), pure extension–transtension (PE–TRN) and transtension (TRN) are examined with the aid of the Win-Tensor stress inversion software. In all of these extensional stress regimes only extensional faults can be activated. The lower dip angle of the reactivated faults is about 40° assuming that the coefficient of friction is no smaller than 0.6. The increase of the stress ratio and/or the fault dip angle up to 70° results in the increase of the slip deviation from the normal activation. Based on the present examination of the slip preference and slip tendency in different extensional stress regimes, a new simple and practical method is proposed herein in order to separate originally heterogeneous fault-slip data into homogeneous fault groups, by which different extensional stress regimes could be determined. The application of the method on the already published fault-slip data of Lemnos Island supports its validity since over 90% the resulted fault groups and stress regimes coincide to the already published ones.

Paleostress field reconstruction in the Oslo region

The geodynamic history of a region is archived in its geologic record which, in turn, may reflect deformation patterns that causally can be related to certain configurations of paleostresses. In the Oslo Region, the exposed geological record ranges from Precambrian high-grade metamorphic rocks through Cambro-Silurian sedimentary rocks to Permo-Carboniferous sedimentary and magmatic rocks, the latter being related to the development of the Oslo rift system. We investigate the kinematics of outcrop-scale faults to derive the diversity of paleostress states responsible for the observed strain. For this purpose, we combine different graphical and numerical approaches to separate heterogeneous fault-slip data sets and estimate the associated reduced stress tensors. A reduced stress tensor consists of the directions of the three principal stress axes with σ 1 ≥ σ 2 ≥ σ 3 and the ratio of principal stress differences, R = ( σ 2 − σ 3)/( σ 1 − σ 3). The present study gives evidence for three major regional paleostress fields that affected the Oslo Region: A compressional stress field with a NW–SE-directed maximum compression ( σ 1) has been identified as a Caledonian imprint. The most prominent regional stress field, however, is tensional, characterised by a WNW–ESE directed minimum compression ( σ 3), and related to the Permo-Carboniferous rifting. Later, the area was affected by a wrench regime with a roughly N–S directed σ 1 and a maximum age of Permian, whereas the absolute timing of this stress field is unclear due to the lack of exposed rocks younger than Permian. For a large number of estimated stress states, none of the principal axes are sub-vertical. These “oblique” paleostresses cannot be integrated into a common regional stress field but rather correspond to local effects of intense magmatic activity during the Permo-Carboniferous phase of rifting. The present study tends to show that the Oslo Region remained unaffected by major tectonic activity for much of the Mesozoic and Cenozoic – which is when the Central European Basin System to the south experienced several phases of intense deformation.

Paleostress states at the south-western margin of the Central European Basin System — Application of fault-slip analysis to unravel a polyphase deformation pattern

We analyse the deformation pattern along the south-western margin of the Central European Basin System (CEBS) where Upper Carboniferous–Mesozoic rocks are uplifted due to the Late Cretaceous basin inversion. The geometry of mesoscale faults and associated striae are used to calculate the stress state(s) responsible for the observed deformation. Each reduced stress tensor obtained comprises (i) the directions of the principal stress axes σ 1, σ 2, and σ 3 ( σ 1 ≥ σ 2 ≥ σ 3) and (ii) the ratio of principal stress differences R = ( σ 2 − σ 3) / ( σ 1 − σ 3). We present a stress inversion technique that allows each stress state inherent in a heterogeneous fault population to be identified by integrating the results of the PBT-Method [Turner, F.J., 1953. Nature and dynamic interpretation of deformation lamellae in calcite of three marbles. American Journal of Sciences, 251(4): 276–298; Sperner, B., Ratschbacher, L. and Ott, R., 1993. Fault-striae analysis: a Turbo Pascal program package for graphical presentation and reduced stress tensor calculation. Computers & Geosciences, 19: 1361–1388] and the Multiple Inverse Method [Yamaji, A., 2000. The multiple inverse method; a new technique to separate stresses from heterogeneous fault-slip data. Journal of Structural Geology, 22(4): 441–452]. This comprehensive approach not only facilitates the separation of complex data sets into homogeneous subsets but also guarantees that each stress state derived fulfils both the criteria of low-misfit angles (Wallace–Bott hypothesis) and high shear-to-normal-stress ratios (Mohr–Coulomb criterion). The reliability of our technique is confirmed by the fact that irrespective of (i) the number of fault-slip data from an outcrop, (ii) the number of subsets they represent and (iii) the proportion of newly formed and reactivated faults, we obtain consistent results from outcrops of variously aged rocks. This consistency concerns both calculated stress states as well as locally observed deformation sequences. Such local chronologies are derived from cross-cutting relationships and superimposition of different fault-slip data which individually are assigned to a consistent stress state. A synthesis of results from different locations of the study area argues for the superposition of two main deformation events. Most prominently, the area was affected by a stress state with a horizontal N–S- to NE–SW-directed maximum compression ( σ 1) and a low stress ratio which induced reverse and strike-slip faulting. A pure strike-slip regime of E–W- to NW–SE-directed compression with moderate stress ratios is less prevalent and probably younger. The age of the youngest rocks documenting these two phases proves the stress fields to correspond to the polyphase Late Cretaceous–Tertiary basin inversion of the CEBS during post-Cenomanian times. The youngest tectonic imprints detected in the study area correspond to locally appearing extensional stress states with varying directions of σ 3. At many sites, the youngest derived paleostress state coincides with the present-day stress field in terms of the direction of the maximum horizontal stress axis, S Hmax. The only stress state detected which pre-dates the basin inversion corresponds to an extensional regime with an WNW-ESE- to NW–SE-directed σ 3-axis observed only locally in the N–S striking Leine Graben and its prolongation to the North. The fact that the majority of variously aged rocks exhibit only the traces of the latest deformation phases indicates a high degree of fault plane reactivation in the study area.

Paleostress analysis of heterogeneous fault-slip data: The Gauss method

We describe the Gauss method for reconstructing paleostress tensors from heterogeneous fault-slip data. We define compatibility measure and compatibility function, which verify the compatibility of a given stress tensor with observed fault-slip data. In order to constrain inversion results to mechanically acceptable solutions, we additionally consider the ratio between the normal and shear stress on the fault plane, since it is assumed that the results of paleostress inversion should be in agreement with the Amonton's Law. The optimal solution for stress tensors that activated the observed faults is found by searching for the global and highest local maxima of the object function F defined as a sum of compatibility functions for all fault-slip data. We verify the reliability of the method both by mathematical means and by numerical tests, and analyse its effectiveness in the case of large dispersion of angular misfit between the direction of slip and shear stress along the faults.