Fault stability analysis plays an important role in assessing the potential hazard of faults and in studying the mechanism of earthquake occurrence. Fault stability depends on the magnitude of the normal and shear stresses imposed on the fault by the tectonic stress and rock friction, while the magnitude of the normal and shear stresses is related to the spatial orientation of the fault normal with respect to the three principal stress axes, so it is easier to understand the variation of fault stability with its orientation by expressing the stability of different faults in the principal axis coordinate system. In this paper, we first developed a method to plot the stability of faults with different orientations in the principal stress axis coordinate system, then investigated the influence of the magnitude of principal stresses and friction on fault instability, and reached the conclusion that the instability is mainly affected by the relative magnitude of principal stresses (shape ratio). Finally, we proposed to use fault stability as an indicator to evaluate the quality of inverted stress obtained from fault slip data or earthquake focal mechanisms, that is, to evaluate the reliability of the inverted stress according to the compatibility of stress and fault stability. It is described in detail in terms of measured fault slip data from two regions.
Hinweise
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Introduction
There are many fissures or faults with different orientations in the interior of the Earth's crust. Because ruptures on existing faults require less energy than ruptures in intact rock, their presence lowers the threshold for earthquakes. Following the assumptions of related studies (e.g., Gephart and Forsyth 1984; Michael 1987; Morris et al. 1996; Vavryčuk 2014) on the intracrustal fault model, this paper assumes that (1) the fault is a regular plane, (2) the entire fault plane behaves in the same way, and (3) reactivation between different faults does not affect each other. Although it has been found that in some cases faults can be partially reactivated or reactivation can jump to different fault systems during rupture (e.g., Cesca et al. 2017; Trippetta et al. 2019), for the pre-rupture phase it can be approximated as a homogeneous rupture. Driven by tectonic stress, not all faults have the opportunity to rupture, but rather faults of certain attitudes can be activated, while the rest of the faults remains dormant all the time. Fault stability is a measure of how difficult it is to activate the fault. If we can reconstruct the underground stress distribution and analyze the stability of faults accordingly, it is possible to explain why earthquakes occur on these faults and help to understand the mechanism of earthquake generation. Currently, there are two main methods for quantitative analysis of fault stability: calculation of the fault instability (Vavryčuk 2014) and slip tendency (Morris et al. 1996). The meaning and principle of these two methods is basically the same, both are based on the friction law to evaluate fault stability: the higher the normal (extrusion) stress or the lower the shear stress acting on a fault, the more difficult it is to activate the fault, they are only slightly different in the way they are calculated. Both methods of fault stability analysis have been widely used in natural geological environments (Collettini and Trippetta 2007; Vavryčuk 2014; Yukutake et al. 2015; Kusumawati et al. 2021) and in industrial mining regions (Moeck et al. 2009; Lei et al. 2020). The stability of faults with different spatial orientations is related to the stress state, including the orientation and magnitude of principal stresses. The distribution of stability among faults is different for different stress states. For a given stress state, the stability of different faults is usually shown by a colored map in the fault strike-dip coordinate system (e.g., Lei et al. 2020; Li et al. 2023) or in the geographic coordinate system (the fault normal is projected onto the equatorial plane by the lower hemisphere equal-area projection) (e.g., Morris et al. 1996; Moeck et al. 2009). This representation has the advantage of visualizing the distribution of different fault stabilities, but the disadvantage is that each figure applies to only one stress state, and it is not easy to see the relationship between the fault stability and its orientation relative to the principal stress axes. Fault stability is calculated from the magnitude of normal and shear stresses, which depends on the spatial orientation of the fault with respect to the principal stress axes. This implies that it is appropriate to plot the fault stability as a function of fault orientation in the principal stress axis coordinate system. In addition, expressing the fault stability in terms of the principal stress axis coordinate can eliminate the dependence of the fault stability on the orientation of the stress itself. In this paper, we have developed a method for displaying the distribution of normal stress, shear stress, and fault stability in the principal stress axis coordinate system, which makes it intuitive to visualize the variation of these three physical quantities with respect to fault orientation relative to the principal stress axes. Varying the values of the parameters that affect the stability calculation also allows one to visualize the extent to which each parameter affects the distribution of fault stability.
In the Earth’s crust, there are latent faults of various orientations, and the activated faults are those with low stability. If we obtain a regional stress state by inverting fault slip data (Angelier 1984; Michael 1984; Rebetsky et al. 2012; Wan 2015), earthquake focal mechanisms (Gephart and Forsyth 1984; Michael 1987; Vavryčuk 2014; Martínez‐Garzón et al. 2016a, Li et al. 2020), or other measures (e.g., Kang et al. 2010; Zhao et al. 2013; Sheng et al. 2021), we can examine the quality of the inverted stresses by forward modeling the stability of the activated faults. If the stability of previously activated faults is low, it means that the stress and the stability of the faults can be adaptive, implying that the retrieved stress is reliable, on the contrary, the accuracy of the retrieved stress should be suspected. In practice, there are many factors that can lead to an incorrect stress. For example, when the stress is derived using fault slip data or focal mechanisms, it is assumed that the fault is ruptured by the background stress, while a fault on which a large earthquake occurs will have a large stress perturbation on the surrounding faults (Stein 1999), and thus the activation of the surrounding faults will be affected by the co-seismic stress released by the large earthquake, which may cause the inverted stress to deviate from the actual situation. In addition, some small earthquakes may occur as a result of localized stress that may be different from the background stress, which is also an unfavorable factor. If there are unfavorable factors that lead to an incorrect stress, the stress will also be incompatible with the stability of the activated faults, which also indicates that the fault stability analysis is valuable for evaluating the reliability of the stress. In this paper, the application of fault stability in evaluating the quality of the inverted stress field is illustrated in detail using the measured fault slip data from two regions as positive and negative examples, TYM (Tymbaki, Crete, Greece) and KAM (Kamogawa, Boso Peninsula, Central Japan), presented in this paper of Angelier (1990).
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Fault stability analysis
If the geological area with uniform stress field is distributed with various kinds of pre-existing faults, depending on the magnitude of normal and shear stresses exerted on the faults by the stress field, the stability or activation difficulty of each fault is different. In simple terms, faults with low normal stress (extrusion nature) and high shear stress are more likely to be activated. Fault instability (Vavryčuk 2014) and slip tendency (Morris et al. 1996) describe the potential for fault activation. In this section, the differences between these two physical quantities and their calculations are presented separately.
Fault instability
The Mohr–Coulomb failure criterion (Eq. 1, Beeler et al. 2000) gives the critical conditions for fault destabilization. The cohesion of an intact, fracture-free rock is very high (e.g., Winn et al. 2017). When an earthquake occurs at the fracture of an intact rock, it takes considerable force to overcome the cohesion of the rock. It is unlikely that the stress in the Earth’s crust can reach this intensity, so it is believed that earthquakes occur at the activation of pre-existing faults, at which time the cohesion of the rock is greatly reduced (Lisle and Srivastava 2004), thereby lowering the threshold for the occurrence of earthquakes. As shown in Fig. 1, under the load of tectonic stress, the radius of the Mohr circle composed of the three principal stresses gradually increases, and when the Mohr–Coulomb failure line is tangent to the Mohr circle composed of \({\sigma }_{1}\) and \({\sigma }_{3}\), the fault reaches the rupture threshold. If there is a fault corresponding to the tangent point in the region, the fault will be the first to be activated. If there is no such fault, the seismic activity in the region is remain calm. With the continuous loading of tectonic stress, the area of the failure line tangent to the Mohr circle increases, the faults within the tangent area satisfy the failure conditions. If these yielding faults are present in the region, they will be destabilized and produce earthquakes. Assuming that the stress in the region is in a critical state, i.e., the failure line is tangent to the Mohr circle, the closer the fault projection in the Mohr circle is to the failure line, the more likely it is to be destabilized. In this regard, Vavryčuk (2014) defines fault instability as Eq. (2) (the significance is the ratio of the length of the blue and red line segments in Fig. 1):
where \(\sigma\) and \(\tau\) are the normal and shear stresses, respectively, \(\mu\) is the friction, P is the pore pressure, and C is rock cohesion:
where \(I\) is the instability, \(I\in \left[0, 1\right]\). \({\sigma }_{c}\) and \({\tau }_{c}\) are the normal and shear stresses on the most unstable faults (red dot in Fig. 1), which can be calculated from the following two equations according to the geometric relation:
where l, m, and k are the cosine angles of the fault normal to the \({\sigma }_{1}\), \({\sigma }_{2}\) and \({\sigma }_{3}\) axes, respectively.
Slip tendency
Earthquakes are ruptures that occur on pre-existing faults, so rock cohesion will be relatively low. Ignoring cohesion, the critical failure condition becomes:
in the equation, \({\sigma }_{e}\) is the effective normal stress.
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Morris et al. (1996) defined the slip tendency as the ratio of the shear stress to the effective normal stress (Eq. 8). It can be normalized by the ratio of the slip tendency to its maximum value (\({T}_{S}/{T}_{{\text{SMAX}}}\)). The greater the slip tendency, the more susceptible the fault is to destabilization:
$$T_{S} = \tau /\sigma_{e} .$$
(8)
Projections of faults in spherical triangle
It can be seen from Eqs. (2) and (8) that fault stability is related to the magnitude of the normal and shear stresses acting on the fault. The magnitude of the normal and shear stresses depends on the square of the cosine of the angles between the fault normal and the three principal stress axes of the stress tensor (Eqs. 5 and 6). Therefore, in the principal stress axis coordinate system, if we draw a sphere with the coordinate origin as the center of the sphere and the unit length as the radius, use the direction from the coordinate origin to each point on the sphere to indicate the normal direction of different faults, and use different colors at each point on the sphere to indicate the stability of different faults, only 1/8 of the sphere is needed to cover all possible faults. We will now consider how to project the 1/8 sphere, which represents the stability of various faults, into the two-dimensional spherical triangle. A similar scenario arises in mapping the classification of earthquake focal mechanism types. Kaverina et al. (1996) used the z-components of the P, B, and T-axis vectors of the focal mechanism as variables to project the focal mechanism onto a spherical triangular focal mechanism classification diagram through equal-area projection, and gave the corresponding projection formulas. In our problem, we refer to its projection, take the absolute values of the fault (unit) normal vector (\(l, m {\text{ and }} k\)) as variables to obtain the position (\(x, y\)) of the fault in the spherical triangle (Eqs. 9–13, see the original paper for details of the derivation):
Distribution of normal stress, shear stress, and fault stability in spherical triangle
In the previous section, we gave a method for projecting a fault into the two-dimensional spherical triangle. By coloring the magnitude of normal stress, shear stress, or stability at each point of the fault projection, the distribution of these physical quantities on variously oriented faults can be obtained. In the following, we will use instability to express fault stability. As it is difficult to directly measure the magnitude of stress in the deep crust, it is generally difficult to give the absolute magnitudes of the three principal stresses. The relative magnitudes of the three principal stresses (denoted by R, Eq. 14), also called shape ratio (Gephart and Forsyth 1984; Michael 1987), can be obtained by inverting fault slip data or focal mechanisms. The R value reflects the sign and stress properties of the intermediate principal stress of the deviatoric stress tensor (Eq. 15) (The sign convention for stress is positive for extrusion and negative for tension.), which represents the different stress modes (Li et al. 2020). When \(R=0\), \({\sigma }_{2}^{\prime}={\sigma }_{1}^{\prime}>0\), at this time, the extrusion stresses of \({\sigma }_{2}^{\prime}\) and \({\sigma }_{1}^{\prime}\) axes are equal, and in this stress mode, the pressures in all directions in the plane formed by \({\sigma }_{1}^{\prime}\) and \({\sigma }_{2}^{\prime}\) axes are equal; when \(0<R<0.5\), \(0<{\sigma }_{2}^{\prime}<{\sigma }_{1}^{\prime}\), \({\sigma }_{2}^{\prime}\) axis is of extrusion nature and the extrusion stress is smaller than \({\sigma }_{1}^{\prime}\) axis. The pressure of \({\sigma }_{2}^{\prime}\) axis decreases with the increase of R; when \(R=0.5\), \({\sigma }_{2}^{\prime}=0\), which means there is no stress on \({\sigma }_{2}^{\prime}\) axis; when \(0.5<R<1\), \({\sigma }_{3}^{\prime}<{\sigma }_{2}^{\prime}<0\), \({\sigma }_{2}^{\prime}\) axis is of tensile nature and the tensile stress is less than \({\sigma }_{3}^{\prime}\) axis. The tensile stress on \({\sigma }_{2}^{\prime}\) axis gradually increases with the increase of R; When \(R=1\), \({\sigma }_{2}^{\prime}={\sigma }_{3}^{\prime}<0\), the tensile stresses in the \({\sigma }_{2}^{\prime}\) and \({\sigma }_{3}^{\prime}\) axes are equal, then the tensile stresses in all directions in the plane formed by the \({\sigma }_{2}^{\prime}\) and \({\sigma }_{3}^{\prime}\) axes are equal in this stress mode. Since the isotropic component of the stress tensor and the magnitude of shear stress do not affect the slip direction of a fault, inverting the fault slip data or the earthquake focal mechanisms yields only a reduced (normalized) stress tensor (Gephart and Forsyth 1984; Michael 1987; Vavryčuk 2014). For calculation purposes, we can follow the same approach as Vavryčuk (2014) to normalize the three principal stresses as \({\sigma }_{1}=1, {\sigma }_{2}=1-2R,{\text{ and }}{\sigma }_{3}=-1\). Correspondingly, under the assumption that the region is in a critical stress state (The failure line is tangent to the Mohr circle at one point), the normalized Mohr–Coulomb failure line equation can be derived and then the instability of various faults can be calculated. Since the instability is defined as the ratio of the length of the line segments (Fig. 1), the instability remains unchanged before and after stress normalization:
where \({\sigma }_{1}\), \({\sigma }_{2}\) and \({\sigma }_{3}\) are the maximum, intermediate and minimum principal stresses, respectively, and \(R\in \left[0, 1\right]\):
where \({\sigma }^{\prime}\) denotes deviatoric stress tensor, \({\sigma }_{1}^{\prime}, {\sigma }_{2}^{\prime} {\text{ and }} {\sigma }_{3}^{\prime}\) are the magnitudes of the principal extrusion, intermediate, and tensile stresses, respectively. \(\mathbf{I}\) is the unit diagonal matrix.
Figure 2 shows the distribution of normal (subfigure a) and shear (subfigure b) stresses over different faults in spherical triangles for \(R=0.5\) and friction to be 0.6. The top, left, and right endpoints of the spherical triangles are the locations of the \({\sigma }_{1}\), \({\sigma }_{2}\) and \({\sigma }_{3}\) axis projections, respectively, and the white arcs indicate the directions with equal angles to the principal stress axes. The normal stress is maximum in the \({\sigma }_{1}\) direction, minimum in the \({\sigma }_{3}\) direction, and 0 in the \({\sigma }_{2}\) direction. The fault normal is close to whichever of 3 principal stress axes the normal stress is closest to the stress of that axis. On the contrary, the shear stress is 0 along the 3 principal stress axes and reaches its maximum when the angle between the fault normal and the \({\sigma }_{1}\) and \({\sigma }_{3}\) axes is about 45°, which corresponds to the highest point of the Mohr circle formed by \({\sigma }_{1}\) and \({\sigma }_{3}\). The instability distribution is shown in Fig. 4c, which shows that a fault with high shear stress and low normal stress has a relatively high instability value. The normal stress, shear stress, and instability all appear to change transiently with the fault orientation. For different stress R values and friction coefficients, the distributions of these three physical quantities are different. We have uploaded the MATLAB program for calculation and drawing to GitHub (website see “Code availability” below), interested readers can try different R values and frictions to observe the changes in the results.
Fig. 2
Distribution of normalized normal (a) and shear (b) stresses for \(R=0.5\). The upper, left and right endpoints of the spherical triangle are the projected locations of the \({\sigma }_{1}\), \({\sigma }_{2}\) and \({\sigma }_{3}\) axes, respectively. The white arcs indicate the direction of equal angles to the principal stress axes. The friction of the drawing is 0.6
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It can be seen that there are two main differences in our mapping approach compared to Morris et al. (1996), who used the lower hemisphere projection to map fault stability, which are also the advantages of the method in this paper. (1) While the mapping method of Morris et al. (1996) is based on the geographic coordinate system, our method is based on the principal stress axis coordinate system, and our method is better able to show the nature of the variation of normal and shear stresses, as well as instability, with fault orientation relative to the 3 principal stress axes. (2) Equations (5) and (6) tell us that only a 1/8 sphere is needed to express the stability of all faults. We plot the information on the 1/8 sphere, while the lower hemisphere projection plot of Morris et al. (1996) plots the information on the 1/2 sphere, so 3/4 of the information is duplicated in their plot.
Influence of friction and magnitude of principal stresses on fault instability distribution
In this section, we will vary the stress R value and the friction to observe the degree of change in the fault instability distribution, thus qualitatively recognizing the effect of these two factors on instability. Figure 3 shows the distribution of fault instability corresponding to different frictions for the stress R value of 0.6. The friction is the slope of the Mohr–Coulomb failure line, we consider a range from 0.4 to 1, which is the range in most geological environments. As can be seen in Fig. 3, the instability distribution varies slightly under different frictions, indicating that friction has less influence on fault instability. We also verified the other R values and came to the same conclusion. Taking friction as an average value of 0.6 within the crust (Jia et al. 2018; Lei et al. 2020), the distribution of fault instability for different R values is shown in Fig. 4. \(R=0\) corresponds to \({\sigma }_{1}={\sigma }_{2}\), so the instability is symmetrically distributed around the \({\sigma }_{3}\) axis (Fig. 4a), and the instability is largest when the angle between the fault normal and the \({\sigma }_{3}\) axis is about 30° (arctan0.6). As the R value increases, the instability distribution in the spherical triangle gradually deviates from symmetry about the \({\sigma }_{3}\) axis. The high instability values are closer to the \({\sigma }_{2}\) axis, and the area of the high values increases, which is due to the decrease in extrusion stress and the increase in tensile stress in the \({\sigma }_{2}\) axis. Until the R value increases to 1, corresponding to \({\sigma }_{2}={\sigma }_{3}\), the instability appears symmetrical about the \({\sigma }_{1}\) axis. The variation of the R value leads to a large fluctuation in the instability distribution, indicating that the R value has a strong influence on the fault instability.
Fig. 3
Distribution of fault instability under different frictions. The upper, left and right endpoints of the spherical triangle are the projected locations of the \({\sigma }_{1}\), \({\sigma }_{2}\) and \({\sigma }_{3}\) axes, respectively. The white arcs indicate the direction of equal angles to the principal stress axes. The labeled black curves are instability contours. The friction of subfigures (a)–(d) is 0.2, 0.4, 0.6, and 0.8, respectively, and the stress R value of each subfigure is 0.6
Fig. 4
Distribution of fault instability for different R values. The upper, left and right endpoints of the spherical triangle are the projected locations of the \({\sigma }_{1}\), \({\sigma }_{2}\) and \({\sigma }_{3}\) axes, respectively. The white arcs indicate the direction of equal angles to the principal stress axes. The labeled black curves are instability contours. The stress R value of subfigures (a)– (e) is 0, 0.25, 0.5, 0.75 and 1, respectively, and the friction of each subfigure is 0.6
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Stress inversion quality assessment based on fault stability
Rationale
The orientation and magnitude (or relative magnitude) of the principal stresses can be obtained from in-situ stress measurements (Barton et al. 1988; Kang et al. 2010; Zhao et al. 2013), fault slips (Angelier 1984; Michael 1984; Rebetsky et al. 2012), or earthquake focal mechanisms (Gephart and Forsyth 1984; Michael 1987; Ghorbani Rostam et al. 2018; Kartal and Kadirioğlu 2019; Kamra et al. 2021), and based on the stress, fault stability can be analyzed. Therefore, the most direct and common application of fault stability is to assess the risk of fault activation. In turn, ruptured faults are necessarily less stable, so the reliability of the stress can be judged by how well the stability of the already destabilized faults matches the stress. Assuming that stress, friction, pore pressure, and cohesion are homogeneous in the regional geologic environment, only faults that satisfy the Mohr–Coulomb failure criterion can rupture. In the \(\sigma -\tau\) coordinate system, faults that satisfy the failure criterion are located within the area of the failure line cutting the Mohr circle. With a known R value, the magnitudes of the three normalized principal stresses can be determined, thereby determining Mohr circle. Friction represents the slope of the failure line. Given friction, normalized pore pressure and cohesion define the position of the failure line in the coordinate system. The values of pore pressure and cohesion may vary in different parts of the crust, and for convenience we do not discuss the specific values of these two physical quantities, but instead discuss the critical instability (minimum instability), which represents the failure condition of faults (given the critical instability, the failure line is also determined). Faults with an instability greater than the critical instability satisfy the failure condition. The range of critical instability is 0 ~ 1. When the failure line is tangent to the Mohr circle at a point on the left, the critical instability is 1. As the failure line moves to the right, the area of the Mohr circle cut by the failure line gradually increases, symbolizing a gradual increase in the variety of faults that satisfy the failure condition. The critical instability will not be extremely high in natural geological environments, since fault slip data or focal mechanisms show that active faults are not uniform, but rather somewhat diverse in their attitude. However, the critical instability will not be relatively low, because earthquakes occur on pre-existing faults with low cohesion, which has little effect on the failure line position, leaving pore pressure as the main factor affecting the failure line position (increasing pore pressure moves the failure line to the right), a lower critical instability would require a large (even exceeding the magnitude of \({\sigma }_{3}\)) pore pressure. The values of critical instability in the previous studies (e.g., Vavryčuk et al. 2015; Martínez-Garzón et al. 2016b) are around 0.8. Referring to their value and our analysis, we can roughly take 0.9, 0.8, and 0.7 as the high, medium, and low levels of the average critical instability within the crust.
Based on the stress inversion result, the cosine value of the angles between the fault normal and the three principal stress axes can be calculated, the expression (\(l, m, k\)) of the fault normal in the principal axis coordinate system can be obtained, and the instability of the faults and their positions in the spherical triangle can be calculated from the theories in “Fault stability analysis” and “Projections of faults in spherical triangle”. The activation of faults follows the failure criterion, so the activated faults should have higher instability. If the instability of the faults is relatively high, then the fault stability is consistent with the stress field, which means that the stress field obtained from the inversion is reliable. On the other hand, if the instability of the faults is relatively low, it indicates that the fault stability does not match with the stress state, thus we have reason to doubt the quality of the data and the stress result. This is the theoretical basis for assessing the stress inversion quality based on fault stability.
Example application
The application of fault stability analysis in evaluating the quality of the stress inversion result will be illustrated using the measured fault slip data given in the paper of Angelier (1990) for two regions, TYM and KAM (Fig. 5). There are a total of 38 faults in TYM and 50 in KAM. The stress states in the TYM and KAM regions were obtained using the linearized inversion method of Michael (1984). The results are shown in Figs. 6, and 7(a–c) and Table 1, where the optimal solution is obtained using the full set of fault slip data, the uncertainty of the stress results in Figs. 6 and 7 are obtained by performing 500 inversions with 80% of the data set randomly selected, and the uncertainty ranges for the 95% confidence intervals are given in Table 1. As can be seen from Figs. 6, and 7 and Table 1, the uncertainty of the stress inversion results in the two regions is relatively small, suggesting that the diversity of the fault slips is able to constrain the stress tensor well. When driven by sufficient stress, a fault will theoretically slip along the shear stress direction (Wallace 1951; Bott 1959), with the actual slip direction being the observed slickenside lineation, so the slip misfit is defined as the angle between the two directions. Subplots c in Figs. 6 and 7 show the statistics of fault slip misfit in the optimal stress scene. The vast majority of the faults in the TYM have slip misfit less than 25°, while a portion of the faults in the KAM have slip misfit greater than 30°, indicating that the stress field in the TYM fits the faults better than that in the KAM.
Fig. 5
Fault slips of TYM (a) and KAM (b). Data taken from Angelier (1990). The projection mode is the lower hemisphere equal-area projection, the red arrows indicate the slip direction of the faults
Fig. 6
Stress results (a and b), slip misfit (c) and variation of average fault instability with friction (d) in the TYM region. Subfigures (a) and (b) show the statistics of the 500 stress inversion results with randomly selected 80% data set, subfigure (a) shows the distribution of stress direction, ‘ + ’ is the optimal stress direction, and subfigure (b) shows the statistics of stress R value, the longitudinal coordinate is the probability density. Subfigure (c) shows the statistics of slip misfit in the optimal stress scene. Subfigure (d) shows the average fault instability for different frictions, the red squares indicate the position of maximum average instability
Fig. 7
Stress results (a and b), slip misfit (c) and variation of average fault instability with friction (d) in the KAM region. The meaning of each subgraph is the same as in Fig. 6
Table 1
Stress inversion results
Region
Number of faults
Stress direction (azimuth/plunge ± uncertainty)
R ± uncertainty
\({\sigma }_{1}\)
\({\sigma }_{2}\)
\({\sigma }_{3}\)
TYM
38
− 132°/84° ± 3°
65°/6° ± 20°
− 25°/2° ± 20°
0.90 ± 0.07
KAM
50
− 139°/84° ± 4°
119°/1° ± 8°
29°/6° ± 8°
0.68 ± 0.08
The optimal stress is obtained using all fault slips, its uncertainty is estimated by performing 500 stress inversions with 80% of the data randomly selected, and the uncertainty ranges with 95% confidence intervals are given in the table
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In “Influence of friction and magnitude of principal stresses on fault instability distribution”, we demonstrate through theoretical calculations that friction has a negligible effect on the instability calculation. To validate our conclusions with real fault slip data and to minimize the effect of friction, we attempted to determine friction using the method in Vavryčuk (2014). This is done by performing a grid search for friction in a range (e.g., 0.4 ~ 1) with a certain step size (e.g., 0.02), calculating the instability of each fault based on the optimal stress for each grid point value of friction, and finally selecting the grid point value that maximizes the average instability of all faults as the final friction value. The variation of the average instability of faults with friction values in the TYM and KAM regions are shown in subplots d in Figs. 6 and 7, respectively, with the red squares denoting the positions of maximum average instability. The average instability of the faults in both regions varies only slightly (on the order of \({10}^{-2}\)) for different friction values, confirming the conclusion that friction has a weak influence on the calculation of fault instability, while suggesting that the friction estimated by the above method is not trustworthy.
Since the effect of friction on the instability calculation is weak, we calculated the fault stability using the friction value estimated from the maximum average instability, in the sense of maximizing the overall instability of the active faults. The distribution of fault instability in the TYM and KAM regions is shown in Fig. 8. The vast majority of faults in the TYM have instability greater than 0.9, and the slip misfit is less than 25° (Fig. 6c), all of which indicate that the inverted stress result is reliable. As a counter example, the overall instability of the faults in the KAM area is relatively low, the faults are widely distributed in the spherical triangles, about 1/5 of the faults have instability lower than 0.7, there is also a part of the faults with slip misfit larger than 30° (Fig. 7c), which indicates that the stress is not well-matched with both the stability of the faults and the slip, the obtained stress result is unreliable. There are many reasons for this phenomenon, such as: (1) faults are activated in different time periods, and the stress state in different time periods is also different; (2) there is a spatial span between the measured faults, and the stress field varies with space; and (3) the partial fault exposed on the surface has a different attitude and slip direction compared with the underground portion of the fault, etc.
Fig. 8
Fault instability in TYM (a) and KAM (b) regions. The friction used to calculate instability is 0.74 in TYM and 0.4 in KAM.
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Conclusions
In this paper, we have done three main researches with the fault stability as the starting perspective, including gave a new method to plot the normal stress, shear stress and stability on various faults under the principal stress axis coordinate system, investigated the extent to which each factor affects the calculation of fault instability, and proposed to use fault stability as a metric for evaluating the quality of stress result from inversion of fault slip data or earthquake focal mechanisms. Some significant results and conclusions were obtained, as detailed below:
(1)
The formulas for calculating the magnitude of normal stress, shear stress, and stability on a fault are all related to the square of the cosine of the angles between the fault normal and the three principal stress axes, which inspired the idea of representing these physical quantities in the principal axis coordinate system. We introduced the method of Kaverina et al. (1996) for projecting earthquake focal mechanism in the spherical triangle to project the fault normal in the principal stress axis coordinate system. With the known magnitude (or relative magnitude) of the principal stresses, the distributions of normal stress, shear stress, and instability were projected with respect to the fault normal in the spherical triangle. This is an important contribution to understanding the distribution patterns of these three physical quantities.
(2)
Assuming a critical stress state, pore pressure and rock cohesion can be determined from the friction coefficient (see section “Fault instability” and Fig. 1), leaving only two factors that affect the calculation of fault instability, the magnitude of the principal stresses and the friction value. We changed the relative magnitude of the principal stresses and the friction value by the control variable method, respectively, to observe the distribution variation of the fault instability in the spherical triangle, to evaluate the influence of these two physical quantities on the instability. It was found that the friction has a weak influence on the distribution of the instability, while the relative magnitude of the principal stresses has a significant influence, indicating that the fault instability is mainly influenced by the magnitude of the principal stresses. Assuming different friction values, the average instability of the measured faults in both the TYM and KAM regions varies only slightly, verifying the conclusion that fault instability is little affected by friction value with actual data.
(3)
we proposed the use of fault stability as an indicator for assessing stress quality obtained from fault slips or focal mechanisms inversion. Under the inverted stress state, if the activated faults have relatively high instability, it implies that the stress and the stability of the faults can be adaptive. Instead, if there is a substantial fraction of faults with lower instability, it indicates that the stress is not conducive to the activation of these faults, which also suggests that the stress result is unreliable. We applied it to evaluate the quality of inverted stress results in the TYM and KAM regions based on measured fault slip data and concluded that the inversion in the TYM region yielded a reliable stress and the inversion in the KAM region yielded an insufficiently reliable stress.
Discussion
There are many physical quantities that measure the stability of faults, such as fracture susceptibility, dilation tendency (Healy and Hicks 2022), etc., in addition to the instability and slip tendency mentioned in this paper. These physical quantities have different definitions and physical meanings. This study analyzed the factors influencing instability and concluded that friction value has little effect on the calculation of fault instability. The conclusion does not apply to other physical quantities. For example, slip tendency is not affected by the friction value, because it is defined independently of friction (Eq. 8), whereas fracture susceptibility can be significantly affected by the friction value (Healy and Hicks 2022).
The principle of using fault slips and earthquake focal mechanisms to invert stress is the same, but the difference lies in the fact that in the process of using earthquake focal mechanisms to invert stress, it is necessary to identify the seismic fault from the two nodal planes provided by the earthquake focal mechanism (Gephart and Forsyth 1984; Vavryčuk 2014). Analyzing the stability of the identified seismic faults based on the recovered stress can also be an indicator for evaluating the quality of the stress inversion. The fit of stress to fault stability and the fit of stress to fault slip direction are both measures of the reliability of the yielded stress. Since only the fault slip direction information is used to construct stress (e.g., Gephart and Forsyth 1984; Michael 1984, 1987; Li et al. 2020), the two metrics are independent, and the fits of the two metrics will not always vary synchronously, suggesting that considering the fit between stress and fault stability is an important and indispensable aspect of assessing stress inversion quality.
Acknowledgements
We would like to thank an anonymous reviewer for thoughtful suggestions that helped improve the manuscript.
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