Implications of Receiver Plane Uncertainty for the Static Stress Triggering Hypothesis
Abstract
Static stress transfer from major earthquakes is commonly invoked as the primary mechanism for triggering aftershocks, but evaluating this mechanism depends on aftershock rupture plane orientations and hypocenter locations, which are often subject to significant observational uncertainty. We evaluate static stress change for an unusually large data set comprising hundreds to thousands of aftershocks following the 1997 Umbria-Marche, 2009 L’Aquila (Italy), and 2019 Ridgecrest (California) earthquake sequences. We compare failure stress resolved on aftershock focal mechanism planes and planes that are optimally oriented (OOPs) in the regional and earthquake perturbed stress field. Like previous studies, we find that failure stress resolved on OOPs overpredicts the percentage (>70%) of triggered aftershocks relative to that predicted from observed aftershock rupture planes (∼50%–65%) from focal mechanisms solutions, independent of how nodal plane ambiguity is resolved. Further, observed aftershock nodal planes appear statistically different from OOPs. Observed rupture planes, at least for larger magnitude events (M > 3), appear to align more closely with pre-existing tectonic structures. The inferred observational uncertainty associated with nodal plane ambiguity, plane orientation, and, to second order, hypocentral location yields a broad range of aftershocks potentially triggered by static stress changes, ranging from slightly better than random chance to nearly any aftershock promoted, particularly those further than 5 km from the causative fault. Dynamic stresses, afterslip, pore fluids, and other sources of unresolved small-scale heterogeneity in the post-mainshock stress field may also contribute appreciably to aftershock occurrence closer to the mainshock.
Key Points
-
Aftershock uncertainties increase the range of statically triggered events from little better than the chance to nearly any event promoted
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Optimally oriented planes for Coulomb failure misrepresent aftershock planes, which may be better represented by considering mapped faults
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Static stress triggering near the mainshock remains uncertain from receiver plane uncertainties, and not only source uncertainties
Plain Language Summary
Large earthquakes are followed by a decaying number of smaller earthquakes, called aftershocks, which are hypothesized to occur on subsidiary faults and fractures stressed by the mainshock. The percentage of aftershocks potentially triggered by this mechanism depends on the locations and orientations of the aftershock faults, and is susceptible to observational uncertainties in these parameters. We evaluate the percentage of aftershocks encouraged by stress changes for the 1997 Umbria-Marche, 2009 L’Aquila (Italy), and 2019 Ridgecrest (California) earthquake sequences using large, published aftershock data sets. While only ∼50%–65% of aftershocks are nominally encouraged by mainshock stress changes, observational uncertainties are large enough that any aftershock could be explained by random chance, or could have been triggered. We also test the common assumption that aftershock faults are oriented to maximize failure in the applied stress field. Contrary to this assumption, we find that aftershocks tend occur on planes oriented more consistently with pre-existing mapped faults. Accounting for aftershock fault plane uncertainties is critical when evaluating the percentage of potentially triggered aftershocks. Additional sources of uncertainty or other triggering mechanisms may be required to explain aftershock occurrence, especially in the immediate vicinity of the mainshock.
1 Introduction
Aftershocks following moderate to large magnitude earthquakes can cause appreciable destruction in regions already affected by larger seismic events despite their relatively small magnitudes. Hence, understanding aftershock occurrence with the goal of maximizing their predictability is crucial for assessing evolving seismic hazard following a major earthquake. Because of the short temporal and spatial associations between mainshocks and their aftershocks, mainshock-induced dynamic and static stress changes have been proposed to explain aftershock occurrence. These coseismic stress changes are potentially enhanced by additional postseismic stress changes arising from poroelastic deformation, afterslip, fluid flow, viscoelastic relaxation, and secondary triggering from preceding aftershocks (e.g., Freed, 2005). However, the relative contribution of each mechanism has proven difficult to assess and remains an open question (e.g., Cattania et al., 2015, 2018; Felzer & Brodsky, 2006; Marsan, 2005; Richards-Dinger et al., 2010). This could, in part, explain why state-of-the-art aftershock forecasting models are primarily empirically based (e.g., epidemic-type aftershock sequence models; Woessner et al., 2011), governed by Omori-law type decay. Physically based forecasts are improving to match the success of empirically based forecasts (e.g., Coulomb rate-and-state models; Cattania et al., 2018; Mancini et al., 2019), but are limited by our lack of knowledge for the rupture planes associated with aftershocks and the relative importance of various aftershock triggering mechanisms, particularly that of static stress transfer.
Coseismic static stress transfer is the most commonly invoked mechanism for aftershock triggering, and has been used to explain the spatial pattern of several different aftershock sequences with varying levels of success (e.g., Freed, 2005; Hardebeck et al., 1998; McCloskey et al., 2003; Nostro et al., 2005; Toda et al., 2012). The mixed success of static stress change as a predictor of aftershock locations stems in part from our lack of knowledge regarding the populations of fault and fractures in the crust that are available for aftershocks to occur on, which must be specified for static stress change calculations. As a result, multiple methods have been implemented for choosing so-called aftershock receiver planes: planes are chosen to maximize failure stress in the regional and earthquake perturbed stress field (i.e., optimally oriented planes [OOPs]; King et al., 1994), chosen retrospectively from aftershock focal mechanism planes (e.g., Nostro et al., 2005), assigned based on knowledge of prior local surface or subsurface structure (e.g., Segou & Parsons, 2020; Steacy et al., 2004), or chosen from some combination of stress-optimum and structurally informed criteria (e.g., Mancini et al., 2020). For example, Nostro et al. (2005) conclude that 82% of the aftershocks associated with the 1997 Umbria-March, Italy earthquake sequence appear triggered by coseismic static stress transfer under the OOP assumption, while only ∼60% would appear triggered if the aftershock nodal planes in higher stress were instead used to infer the aftershock fault plane orientations. In contrast, McCloskey et al. (2003) found ∼89% of aftershocks could have been triggered by static stresses if the planes were constrained to vertically dipping structures with strikes in a range defined by observed surface structure but chosen to maximize the failure stress for the 1992 Landers, California earthquake. McCloskey et al. (2003) further showed that the percentage of triggered aftershocks dropped to ∼79% when assuming purely OOPs. Similarly varied results are reported in numerous additional studies (e.g., Asayesh et al., 2020; Brunsvik et al., 2021).
Another limitation on complete evaluation of the static stress triggering hypothesis is the recognition that heterogeneities exist at a scale below the resolution of existing geophysical data sets. For example, variations in mainshock slip distributions, roughness of the rupture planes, or differences in fault friction, rock permeability, or other physical properties are likely to exist at a scale invisible to state of the art seismological and geodetic methods. Our best models for crustal stress necessarily oversimplify the true variability of the stress field, particularly in the immediate vicinity of the mainshock ruptures where deficiencies in the mainshock source representation are most consequential. In fact, many previous investigators explicitly exclude (a usually larger number of) aftershocks located within about 5 km of the mainshock rupture to hedge against this known limitation (e.g., Hardebeck et al., 1998; King & Cocco, 2001; McCloskey et al., 2003; Steacy et al., 2004; Wang et al., 2014). Several past studies have explored some of the uncertainties associated with the mainshock source (e.g., Hardebeck et al.,1998; Steacy et al., 2004; Wang et al., 2014; Woessner et al., 2012). For example, Brunsvik et al. (2021), infers the roughness of the 2009 L’Aquila mainshock rupture surface by spatial clustering of aftershock data, but finds that the resulting curved fault model results in fewer aftershocks encouraged by static stress change than the planar source model. These varied methodologies and conclusions drawn from past analyses of the static stress triggering hypothesis hamper community consensus for the validity and utility of static stress as a predictor of aftershock locations. Here, we complement these previous studies by focusing on an evaluation of receiver fault plane uncertainty in both the near and far-field of the rupture planes.
Retrospective analysis using rupture planes from observed aftershock focal mechanism solutions should provide the most realistic gauge for assessing static stress transfer from large magnitude earthquakes as a primary aftershock triggering mechanism; however, focal mechanism data carry the ambiguity of which nodal plane represents the true rupture plane at depth. Random vs. informed choices of nodal planes produce statistically significant differences in the evaluation of static stress triggering (e.g., Hardebeck et al., 1998; Nandan et al., 2016; Ziv & Rubin, 2000). Nandan et al. (2016) choose nodal planes by fitting Gaussian mixture models to groups of events clustered by a measure of waveform similarity. More simply, the question of nodal plane ambiguity is often resolved based on a comparison with mapped faults, motivated by observations that aftershocks tend to align well with such structures (McCloskey et al., 2003; Segou & Parsons, 2020; Steacy et al., 2004; Ziv & Rubin, 2000). Both methods support the static stress triggering hypothesis better after resolving nodal plane ambiguity. In regions with complex structure, or for small magnitude events that arguably are less likely to match large-scale mapped structure, choosing the correct nodal plane becomes more challenging (e.g., Nostro et al., 2005); however, it is important to address for a discussion of mainshock-induced static stress triggering.
Although not directly related to static stress transfer, nodal plane ambiguity has been addressed for the purpose of regional or local stress field inversions via an instability constraint (Vavrycuk, 2014), which may also be applicable to choosing aftershock fault planes for analysis of stress triggering. For stress inversion studies, the more unstable (i.e., more likely to fail) nodal plane in the assumed uniform regional stress field is selected. This method has achieved high levels of success for picking fault planes in data sets for which the local structure is independently constrained (Vavrycuk, 2014). Hardebeck (2020) further demonstrates that aftershocks are more likely to occur in regions where the earthquake stress field aligns with the background regional stress field, implying that already critically stressed faults in the regional stress field are encouraged toward failure where earthquake-induced stresses constructively add to the regional stress field. It may follow that the more unstable nodal plane in the regional and earthquake perturbed stress field, or simply the regional stress field, is a method for resolving nodal plane ambiguity. This is something we test.
Outside of the issue of nodal plane ambiguity, the focal mechanisms themselves are prone to observational errors that have not been thoroughly investigated in terms of static stress transfer for real aftershock focal mechanism data sets. Wang et al. (2014) demonstrate that static stress triggering calculations are particularly sensitive to the dip of receiver planes when compared to uncertainties introduced by different mainshock slip distributions and the effective friction coefficient, strikes, and rakes of receiver planes. Their analysis evaluates this sensitivity at aftershock hypocentral locations, but assumes receiver fault orientations and uncertainties consistent with the mainshock focal mechanism, and does not consider individually observed aftershock nodal plane orientations. Ziv and Rubin (2000) find that the reported strike, dip, rake, and location errors associated with focal mechanisms for 63 California earthquakes has little effect on the percentage of aftershocks promoted by static stress change, but the data set is small, including only relatively large magnitude (M > 4.5), well-constrained events. Hainzl et al. (2010) demonstrate the importance of incorporating a distribution of receiver fault strikes in Coulomb rate-and-state models to better predict aftershocks observed after the 1992 Landers, California earthquake, but does not test distributions for the actual observed aftershock nodal planes. No thorough investigation using focal mechanism uncertainties for individual aftershocks to assess the static stress triggering hypothesis exists, though static stress calculations are sensitive to these parameters.
The aforementioned issues lead us to pose two main questions regarding the static stress triggering hypothesis: (a) Can coseismic static stress change from the largest magnitude events truly be discounted as a triggering mechanism for any or all aftershocks if we consider location and orientation uncertainties for observed aftershock nodal planes? (b) Is there a preferred method for resolving nodal plane ambiguity in the regional and earthquake perturbed stress field, and can we gain a better understanding of the structure and spatial distribution of statically triggered or untriggered aftershocks? In this study, we explore a range of failure stress calculations applied to aftershock data sets for the 1997 Umbria Marche and 2009 L’Aquila (Italy) normal faulting sequences, and the 2019 Ridgecrest (California) strike-slip faulting sequence to retrospectively evaluate the potential of mainshock-induced static stress change for triggering aftershocks (Figure 1). We also resolve the stress changes on OOPs for comparison with the aftershock nodal plane results, and in the process re-evaluate the range of potentially triggered aftershocks under different approaches for resolving nodal plane ambiguity. We compare these results to random synthetic aftershock sequence tests. The nearly unprecedented number of focal mechanism solutions for these events' aftershock sequences allows us to extend the evaluation of uncertainty for coseismic static stress transfer in triggering aftershocks, complementing previous studies. The available data sets also provide an opportunity to assess the structure illuminated by the activated fault planes during each complex earthquake sequence in varied tectonic environments.
(left) Map of the 2019 Ridgecrest, California earthquake sequence study area. Aftershocks analyzed (maroon dots) are from the Lin (2020) catalog. The bolded surface traces of the modeled coseismic slip distributions for the foreshock and mainshock are from Ross et al. (2019). Mapped faults from the USGS quaternary fault database prior to the Ridgecrest rupture are presented as thin black traces, and moment tensors are from the Caltech/USGS Southern California Seismic Network (1926). (right) Map of the northern-central Apennines, Italy with data for the 1997 Umbria-Marche and 2009 L’Aquila earthquake sequences. Aftershocks used in this study for the Umbria-Marche sequence are from Chiaraluce et al. (2003), with coseismic slip distributions from Hernandez et al. (2004) outlined as bold black lines, and moment tensors from Ekström et al. (1998). Aftershocks for the L’Aquila earthquake sequence are from Chiaraluce et al. (2021), with the mainshock slip distribution of Cirella et al. (2012) outlined in black, and the moment tensor from Scognamiglio et al. (2009).
2 Data and Methods
In this section, we describe the models and data relevant to evaluating static stress change resolved on a range of planes at aftershock hypocenters for each earthquake sequence, then describe our method for incorporating strike, dip, rake, and location uncertainties of focal mechanisms in the calculations, and finally describe our workflow to evaluate whether the nominal results for static triggering surpass that of random synthetic tests.
2.1 Coseismic Slip Distributions, Aftershock Data Sets, and Regional Stress
We calculate the coseismic static stress change tensor at the hypocenters of aftershocks using the publicly available program RELAX 1.0.7 (Barbot, 2014; Barbot & Fialko, 2010a, 2010b), published at the Computational Infrastructure for Geodynamics under the GPL3 license. RELAX evaluates Green's functions for dislocation sources in an elastic half-space under gravity in the Fourier-domain, with coseismic and postseismic processes represented by equivalent body-forces. We assume an elastic half-space with Lame's parameters calculated from crustal velocities reported in crust1.0 (Laske et al., 2013), resulting in a lambda of 32.9 GPa and shear modulus of 34.5 GPa. These values are the same for both Italy and California. Stress calculations in the model domain are numerically evaluated at a specified grid spacing, with the minimum advisable spacing depending on the spatial resolution of the input slip distributions. The edges of the domain are greater than five fault lengths from the rupture planes to reduce numerical error associated with elastic solutions calculated in the frequency domain. The inputs provided at runtime are available on GitHub (https://zenodo.org/record/6289582).
The best available coseismic slip distributions for each of the three earthquake sequences are modeled as an arrangement of rectangular uniform slip patches and described in the following sections. In all cases, we choose to model stress change arising from only large earthquakes for which heterogeneous slip distributions have been reported. Our preliminary work revealed little change in the percentage of triggered aftershocks for the Umbria-Marche sequence when smaller magnitude events were modeled with uniform slip patches according to earthquake scaling relationships (Hanks & Bakun, 2002). For the L’Aquila earthquake sequence, incorporation of smaller magnitude events as uniform slip patches decreased the percentage of triggered aftershocks. Brunsvik et al. (2021) noted similar findings for the L’Aquila earthquake sequence. Meier et al. (2014) also concluded that incorporating smaller magnitude events with earthquake scaling relationships introduced too much uncertainty for stress calculations. We, therefore, omit events of magnitude <5.5 without reported slip distributions to avoid incorporating further uncertainty in our models of coseismic static stress change, and highlight that the goal of this study is not to investigate secondary triggering from smaller magnitude events.
2.1.1 The Umbria-Marche Earthquake Sequence
We model the coseismic static stress changes associated with the three largest earthquakes of the 1997 Umbria-Marche sequence (Mw 5.7 26 September 1997, Mw 6.0 26 September 1997, Mw 5.6 14 October 1997) with published slip distributions (Hernandez et al., 2004). The slip distributions were estimated using GPS, Differential Interferometric Synthetic Aperture Radar (DInSAR), and strong motion data (Figure 1; Hernandez et al., 2004). These slip distributions are consistent with those used in the similar study of Nostro et al. (2005), though we model only the three largest events, as opposed to their eight, which included smaller magnitude events without detailed slip distributions.
The aftershock data set includes 320 focal mechanism solutions from first motion polarity data (Figure 1; Chiaraluce et al., 2003). The data set begins on 26 September 1997 following the first two modeled events, and ends on 28 October 1997 with events in the magnitude range of Md 2.4–5.5. This data set is notably an order of magnitude smaller than for L’Aquila or Ridgecrest, but all data sets have a similar resolution. Because different methods were used to estimate magnitudes in the aftershock data sets, we use M to denote various magnitude types. The standard deviations in uncertainty for aftershock hypocenter locations are ±70, 85, and 120 m in latitude, longitude, and depth, respectively (Chiaraluce et al., 2003). Standard deviations representing uncertainty in strike, dip, and rake are reported at ∼10°, 15°, and 30°, respectively (Table 1; Chiaraluce et al., 2004).
| Earthquake | Latitude (m) | Longitude (m) | Depth (m) | Strike/dip/rake (°) |
|---|---|---|---|---|
| Umbria-Marche | 70 | 85 | 120 | 10/15/30 |
| L’Aquila | 39 | 178 | 87 | 10/15/30 |
| Ridgecrest | 75 | 75 | 108 | 10/15/30 |
The deviatoric regional stress tensor required to compute OOPs and assess nodal plane instability is derived from a focal mechanism inversion of the six largest earthquakes in the sequence (Chiaraluce et al., 2003), with a nearly vertical maximum compressional principal stress (compression is taken here as negative) of −2.0 MPa (
), the axis of minimum compressional principal stress (
) trending 53°, and a stress ratio (
) of 0.6, leading to a maximum differential stress of 4.3 MPa. The magnitude is consistent with stress drops of the largest earthquakes in the region, as chosen in Nostro et al. (2005), but also considers the stress ratio for the relative magnitudes of the deviatoric principal stresses.
2.1.2 The L’Aquila Earthquake Sequence
We model the coseismic static stress change from the largest magnitude event of the L’Aquila earthquake sequence (Mw 6.1 6 April 2009), with a slip distribution determined from a combination of GPS, DInSAR, and strong motion data (Figure 1; Cirella et al., 2012).
The aftershock data set includes 3,415 focal mechanism solutions compiled from multiple sources (Figure 1; Chiaraluce et al., 2021). The data set begins following the 6 April 2009 mainshock, and ends on 2 December 2009, with events in the magnitude range of Ml 0–5.4. The hypocentral location standard deviations are ±178, ±39, and ±87 m in the two horizontals and in depth, respectively (Valoroso et al., 2013). As for Ridgecrest, in absence of estimated uncertainties for these parameters, we assume the standard deviations in strike, dip, and rake are consistent with the values for the Umbria-Marche data set (10° strike, 15° dip, and 30° rake; Table 1).
The regional stress tensor is assumed to be the same as for the Umbria-Marche region to calculate OOPs for failure. Comparison to the principal stress directions available via the World Stress Map (Heidbach et al., 2018) closer to the L’Aquila earthquake supports this assumption.
2.1.3 The Ridgecrest Earthquake Sequence
We model coseismic static stress changes for the two largest events of the Ridgecrest earthquake sequence (Mw 6.4 4 July 2019, Mw 7.1 6 July 2019). The adopted slip distributions were determined from GPS and InSAR displacements (Figure 1; Ross et al., 2019).
The aftershock data set consists of 2,888 focal mechanisms from first motion polarity data, using focal mechanisms from the Lin (2020) data set (Figure 1). The aftershock data set begins following the foreshock on 4 July 2019 and ends on 31 August 2020 with events in the magnitude range of M 0.9–5.4. The location standard deviations are ±75 m in the horizontal and ±108 m in the vertical (Lin, 2020). Although the fault plane orientation uncertainties vary as reported in the original data set, we choose again to assume standard deviations of 10° in strike, 15° in dip, and 30° in rake, because the reported uncertainties, not distinguishing between the three parameters, fall around 30° for overall plane orientation (Table 1).
We apply a uniform deviatoric regional stress tensor from the closest point to the 6 July mainshock of Luttrell and Smith-Konter (2017), available through the Southern California Community Stress Model (https://www.scec.org/research/csm), to calculate OOPs for failure. The maximum compressional deviatoric principal stress has a value of −2.1 MPa, with an azimuth of 192° and a plunge of 36°. The differential stress is 6.2 MPa.
2.2 Modeling Coulomb Failure Stress for Aftershocks
) plus the change in resolved normal stress (
; with compression as negative) weighted by the effective coefficient of friction (
) for the receiver fault plane (e.g., King et al., 1994):
The effective coefficient of friction is the nominal value of friction for the fault, modified to account for the reduction of normal stress from pore fluid pressure (e.g., Cocco & Rice, 2002). The effect of varying 
on CFS calculations has been thoroughly investigated elsewhere, and is found to have a significant but secondary effect in the calculated results (e.g., Hardebeck et al., 1998; Steacy et al., 2004; Wang et al., 2014). Because we focus on the range of permissible ∆CFS from receiver plane location and orientation uncertainties alone, we choose a common value for 
of 0.6 for all calculations (e.g., Brace & Byerlee, 1966). Failure of an aftershock plane is considered to be promoted for positive ∆CFS, and inhibited for negative ∆CFS. We assume no threshold for triggering exists, consistent with the findings of Ziv and Rubin (2000). Also, we use ∆CFS to denote the modeled coseismic stress changes in this article, while CFS indicates superposition with the deviatoric regional stress field.
2.3 Optimally Oriented Planes (OOPs)
), derived from the combined regional (
) and earthquake perturbed (
) stress fields (King & Cocco, 2001; King et al., 1994):
The two OOPs are those for which CFS is maximized in the total stress field, in the direction of maximum slip for that plane. This calculation is performed at the hypocenters of all available aftershocks. Note that the plane and slip orientations are calculated in the total stress field, but the ∆CFS values reported here include only static stress change associated with modeled coseismic displacements.
2.4 Addressing Nodal Plane Ambiguity
Each focal mechanism consists of two nodal planes, leaving the question of which plane represents the true fault plane at depth. Therefore, we evaluate several methods for resolving this ambiguity, and present results for multiple plane categories defined as follows: (I) The nodal planes in lower ∆CFS (Lower ∆CFS); (II) The nodal planes in higher ∆CFS, considered the most encouraged toward failure in the earthquake-induced stress change field (Higher ∆CFS); (III) The nodal planes that are most encouraged toward failure in the combined deviatoric regional and earthquake perturbed stress field (Reg + Eq Unstable); (IV) The nodal planes that are most encouraged toward failure in the deviatoric regional stress field alone (Reg Unstable); (V) The nodal planes that align best with local mapped fault traces (Strike Constrained). These categories are summarized in Table 2.
| Plane category name | Description | |
|---|---|---|
| OOP | Optimally oriented planes | Conjugate planes calculated following Section 2.3 |
| I | Lower ∆CFS | Nodal plane in lower coseismic ∆CFS |
| II | Higher ∆CFS | Nodal plane in higher coseismic ∆CFS |
| III | Reg + Eq unstable | Nodal plane in higher CFS in the regional and coseismic stress field |
| IV | Reg Unstable | Nodal plane in higher CFS in the regional stress field |
| V | Strike Constrained | Nodal plane best aligned with the strike of mapped faults |
The strike constraint is determined from mapped faults in each study region (Figure 1). For the regions hosting the Ridgecrest and L’Aquila earthquake sequences, mapped surface fault strikes include two peak directions (Pizzi & Galadini, 2009; USGS Qfaults database pictured in Figure 1). For Ridgecrest, Strike Constrained planes (V) are those that align best with strikes of 340°/160° and 80°/260° (USGS Qfaults database pictured in Figure 1). For L’Aquila, Strike Constrained planes (V) are those that align best with strikes of 305°/125° and 75°/255° (Pizzi & Galadini, 2009). For Umbria-Marche, Strike Constrained planes (V) are those that align best with the single strike of 145°/325° (Pizzi & Galadini, 2009).
In the next section, we develop statistical descriptions of each nodal plane, then apply the category criteria of Table 2 to restrict the set of planes such that each aftershock has only one nodal plane.
2.5 Modeling the Distributions of ∆CFS
The parameters describing each nodal plane (strike, dip, rake, and three location coordinates) denote nominal estimates subject to observational uncertainty. To account for this uncertainty, we treat the parameters with mean values equal to the nominal parameter estimates, and variances equal to the square of the assumed error standard deviations listed in Table 1. Under this assumption, the joint probability distribution for each set of orientation and position parameters is the product of the marginal distributions representing each of the nodal plane's parameter estimates. This assumption is violated in the sense that for any given aftershock focal mechanism, the strike, dip, and rake of one nodal plane are 100% correlated with the strike, dip, and rake of its corresponding auxiliary nodal plane. However, this issue does not complicate our statistical analysis because for any given category in Table 2, we will be concerned with only one of the two nodal planes per aftershock, as mentioned in the previous paragraph.
We approximate the distributions of ∆CFS for each aftershock by random sampling from the joint nodal plane parameter distributions and resolving the stress tensor field (Section 2.1) onto the sampled planes at the sampled locations according to the following procedure: First, the coseismic stress change is calculated at each sample location by interpolating the stress tensor field via a first order Taylor series expansion about the nominal aftershock location. The spatial gradients of stress components needed for the series expansion are computed from the grid of nodes defining the model domain (cf. Section 2.1). The interpolation introduces only minor changes compared to the receiver plane uncertainty. Stress tensors are evaluated for each aftershock at 100 sampled locations and then, for each of the 100 locations, 50 random combinations of strike, dip, and rake are sampled. This yields 5,000 ∆CFS samples for each nodal plane. Finally, distributions of ∆CFS associated with each nodal plane are approximated from histograms of the 5,000 ∆CFS values computed for that nodal plane (Figure 2).
Example ∆CFS distribution for one nodal plane of an aftershock focal mechanism, containing 5,000 ∆CFS values calculated using combinations of position, strike, dip, and rake sampled from normal distributions varied about the nominal parameter values with variances equal to the squared errors reported in Table 1. Percentile values are used to assess the probability that the rupture plane of this aftershock experienced positive ∆CFS. For this example, the 18th percentile splits the distribution between positive and negative ∆CFS (black dashed line), meaning 82% of the sampled planes for this aftershock experienced positive ∆CFS. We interpret this as a measure of probability that the true rupture plane experienced positive ∆CFS. The 16th/84th and 2nd/98th percentile values are used in the full aftershock analysis to calculate percentage ranges of potentially triggered aftershocks (Figure 3).
We use percentile values from each planes' distribution to assess the probability that the rupture plane experienced positive ∆CFS, illustrated in Figure 2. While the ∆CFS value for the nominal aftershock plane is a useful metric, we use the 16th and 84th percentile values of each individual aftershock distribution to define a 1σ percentage range of promoted aftershocks for the whole aftershock sequence (Figure 3). The lower bound of the 1σ percentage range is defined as the percentage of aftershock nodal planes that experience positive ∆CFS at the 16th percentile value of their respective distributions, and the upper bound is defined as the percentage of all aftershocks that experience positive ∆CFS at the 84th percentile value of their distributions. Similarly, the 2σ percentage range is bounded on the lower end by the percentage of aftershock nodal planes with 2nd percentile values in positive ∆CFS, and bounded on the upper end by aftershock nodal planes with 98th percentile values in positive ∆CFS.
Percentages of aftershocks in positive ∆CFS for OOPs (red bars) and categories of planes as described in the text. The black bars indicate the percentages of promoted aftershocks with the nominal nodal plane orientations and locations. The darker colored regions encompass the 1σ percentage range after incorporating orientation and location uncertainties, bounded on the lower end by the percentage of aftershocks with a positive 16th percentile ∆CFS distribution value, and an upper bound defined by the percentage of aftershocks with a positive 84th percentile ∆CFS distribution value. The lighter colored regions are the 2σ percentage range, similarly defined but at the 2nd and 98th percentiles. Table 3 contains these percentage results.
2.6 Synthetic Aftershock Sequences
We perform three synthetic tests with ∼5,000 aftershocks for each earthquake sequence to verify that the observed percentage of statically triggered events surpasses that of a random test with the same distribution of planes. We compute the distances of the observed aftershocks to the rupture planes, isolate the aftershocks occurring closest to the ruptures by kmeans clustering, and compute the standard deviations of those clusters (Table S1 in Supporting Information S1). For Umbria-Marche, there is no distinct change in the number of aftershocks with distance from the planes, so we use the full data set for the distance standard deviation. We use the standard deviations to compute normal distributions of 4,500 random points centered on the rupture planes, add background points that are randomly but uniformly distributed to a depth of 10 km to represent observed seismicity that occurs further from the rupture planes, then remove points outside of the model domain. Strike, dip, and rake orientations are randomly assigned to each point from the Strike Constrained (V) category of nodal planes, noting that there is little observable difference in the strike of the planes with distance from the rupture or timing during the earthquake sequence. For Ridgecrest and Umbria-Marche with more than one modeled rupture, the timing of the stress changes for the aftershocks are assigned based on which rupture plane distribution they are associated with, or randomly assigned in the background group with a percentage of events following each rupture that approximates the observed sequences.
3 Results
We present the percentage of aftershocks whose sample distributions have positive ∆CFS at given percentile values for: (a) The OOPs computed from the modeled regional and earthquake perturbed stress field; (b) The nodal plane categories selected according to the criteria described above (I: Lower ∆CFS; II: Higher ∆CFS; III: Reg + Eq Unstable; IV: Reg Unstable; V: Strike Constrained; Table 2); (c) The overall results for each sequence considering both focal planes, without resolving nodal plane ambiguity. Table 3 and Figure 3 summarize the results. We then compare the OOPs to the observed nodal plane orientations and slip directions for each aftershock sequence. Finally, we compare the results to the synthetic aftershock sequence tests.
| Umbria-Marche | L’Aquila | Ridgecrest | |
|---|---|---|---|
| OOP | 71.2 | 85.0 | 96.6 |
| Plane normal orientation difference (mean) | 24.5° | 26.0° | 33.8° |
| Slip vector orientation difference (mean) | 35.1° | 37.7° | 45.6° |
| All focal planes | 46.4 | 69.4 | 66.8 |
| 1σ | 15.8–63.5 | 25.9–86.9 | 22.9–82.0 |
| 2σ | 3.0–81.8 | 16.0–97.7 | 11.0–93.3 |
Higher  |
60.2 | 79.2 | 78.3 |
| 1σ | 39.2–76.2 | 60.3–92.7 | 61.5–90.7 |
| 2σ | 9.1–89.3 | 37.7–99.3 | 34.8–97.1 |
| Strike Constrained | 50.5 | 65.8 | 65.1 |
| 1σ | 31.0–68.0 | 46.6–86.2 | 46.2–80.3 |
| 2σ | 7.2–84.0 | 26.9–97.9 | 22.4–93.5 |
| Reg Unstable | 49.5 | 67.6 | 68.0 |
| 1σ | 29.8–62.7 | 49.6–86.1 | 49.8–81.3 |
| 2σ | 7.5–82.1 | 30.2–97.7 | 26.2–92.3 |
| Reg + Eq Unstable | 54.9 | 72.5 | 77.0 |
| 1σ | 33.2–69.9 | 52.4–89.8 | 59.4–89.3 |
| 2σ | 8.1–85.9 | 31.0–98.7 | 32.5–96.6 |
Lower  |
32.6 | 57.5 | 55.4 |
| 1σ | 13.8–50.8 | 37.4–81.0 | 35.9–73.2 |
| 2σ | 0.9–74.3 | 17.0–96.0 | 14.5–89.4 |
- Note. The 1σ and 2σ percentage ranges correspond to the percentage of aftershocks with positive ∆CFS for the 16th and 84th, and 2nd and 98th percentiles of their ∆CFS distributions, respectively.
3.1 Optimally Oriented Planes
For the three earthquake sequences, OOPs yield the largest percentage of aftershocks in positive ∆CFS, not considering the ranges after accounting for focal mechanism plane uncertainties (Table 3; Figure 3). For Umbria-Marche, 71.2% of aftershocks are in positive ∆CFS from the modeled mainshocks. The percentage for L’Aquila is higher, with 85.0% in positive ∆CFS, and the highest for Ridgecrest, with 96.6% of aftershocks in positive ∆CFS.
The aftershocks in positive ∆CFS are scattered in space, while the unpromoted aftershocks generally fall near the modeled coseismic rupture planes (Figure 4). For the Umbria-Marche sequence, the OOPs in negative ∆CFS are mixed with others in positive ∆CFS, generally near the rupture planes and in the hanging wall of the three mainshocks. Many of the OOPs in negative ∆CFS for the L’Aquila earthquake sequence are within a few km of the mainshock rupture plane in the hanging wall, though two distinct groups lie in the footwall several km to the north near a depth of ∼8 km. For Ridgecrest, many of the events in negative ∆CFS lie to the southeast of the Mw 6.4 foreshock rupture plane, with one cluster to the northwest of the Mw 7.1 mainshock, close to the rupture plane.
Aftershocks in positive ∆CFS (magenta diamonds) and negative ∆CFS (green diamonds) for OOPs and Strike Constrained (V) planes for each earthquake sequence. The backgrounds are color gridded by ∆CFS resolved on OOPs and randomly selected Strike Constrained (V) planes using the static stress change field at 8 km depth. Yellow stars mark the hypocenters of the modeled mainshocks. Aftershocks, slip distribution outlines, and fault traces are the same as in Figure 1. For a view of these aftershocks in 3D, with options for viewing ∆CFS at different percentiles, visit https://observablehq.com/@cehanagan/aftershock_stress.
3.2 Focal Mechanism Planes
In the following subsections, we present the percentages of aftershocks in positive ∆CFS for the given nodal plane categories (I: Lower ∆CFS; II: Higher ∆CFS; III: Reg + Eq Unstable; IV: Reg Unstable; V: Strike Constrained; Table 2). Figure S1 in Supporting Information S1 shows map views of aftershocks in positive and negative ∆CFS for each plane category, except for the Strike Constrained (V) planes, which are presented in Figure 4. We focus many of our results and discussion on the Strike Constrained (V) planes because, as explained in Section 3.4 and in the discussion, this is our preferred nodal plane category. These results may be viewed interactively in 3D at https://observablehq.com/@cehanagan/aftershock_stress.
3.2.1 Umbria-Marche
Of the analyzed earthquake sequences, the lowest percentage of aftershocks are in positive ∆CFS for Umbria-Marche, but the spread of the results is the largest between focal mechanism plane categories (Figure 3). Considering both nodal planes, the average percentage in positive ∆CFS is 46.4% (1σ: 15.8%–63.5%; 2σ: 3.0%–81.8%). The maximum percentage of triggered aftershocks, based on the plane in Higher ∆CFS (II) (i.e., most unstable in the earthquake perturbed stress field) is 60.2% (1σ: 39.2%–76.2%; 2σ: 9.1%–89.3%). The percentage of aftershocks in positive ∆CFS for the Reg + Eq Unstable (III) planes is expectedly lower at 54.9% (1σ: 33.2%–69.9%; 2σ: 8.1%–85.9%), because unstable planes in the regional stress field do not necessarily coincide with planes in higher ∆CFS from the coseismic stress change field. The percentage of aftershocks in positive ∆CFS considering the Eq Unstable (IV) planes is 49.5% (1σ: 29.8%–62.7%; 2σ: 7.5%–82.1%). Similarly, the Strike Constrained (V) planes find 50.5% (1σ: 31.0%–68.0%; 2σ: 7.2%–84.0%) in positive ∆CFS, higher than the average of both planes. The minimum percentage of promoted aftershocks, as determined from the plane in Lower ∆CFS (I), is 32.6% (1σ: 13.8%–50.8%; 2σ: 0.9%–74.3%).
The spatial distribution of planes in positive and negative ∆CFS is similar, regardless of the method applied for choosing focal mechanism planes. The aftershocks in positive ∆CFS are generally intermixed with the events in negative ∆CFS, though a notable number in the hangingwall of the large events are in negative ∆CFS when compared to the other earthquake sequences.
3.2.2 L’Aquila
Considering both focal mechanism planes for L’Aquila, the average percentage in positive ∆CFS is 69.4% (1σ: 25.9%–86.9%; 2σ: 16.0%–97.7%). The maximum percentage of triggered aftershocks, considering the Higher ∆CFS (II) planes, is considerably higher at 79.2% (1σ: 60.3%–92.7%; 2σ: 37.7%–99.3%). The percentage of aftershocks in positive ∆CFS for the Reg + Eq Unstable (III) planes is 72.5% (1σ: 52.4%–89.8%; 2σ: 31.0%–98.7%), while the percentage of aftershocks in positive ∆CFS for the Reg Unstable (IV) planes drops to 67.6% (1σ: 49.6%–86.1%; 2σ: 30.2%–97.7%). The Strike Constrained (V) planes find a slightly lower percentage of 65.8% (1σ: 46.6%–86.2%; 2σ: 26.9%–97.9%) in positive ∆CFS, notably less than the average for both planes. The minimum percentage of promoted aftershocks from the plane in lower ∆CFS is 57.5% (1σ: 37.4%–81.0%; 2σ: 17.0%–96.0%). The percentage ranges for L’Aquila extend the upper bound on the percentage of potentially triggered aftershocks to within a few percent of 100 for any category.
As with the Umbria-Marche sequence, there is little difference in the spatial distribution of aftershocks in positive or negative ∆CFS for different nodal plane categories. More aftershocks in negative ∆CFS extend to the north when compared to the OOPs, primarily associated with the listric Campotosto fault, which hosted 3 large magnitudes (M ≥ 5) aftershocks (Figure 4; Cheloni et al., 2014; Valoroso et al., 2013).
3.2.3 Ridgecrest
For Ridgecrest, the average percentage of aftershocks in positive ∆CFS for both planes is 66.8% (1σ: 22.9%–82.0%; 2σ: 11.0%–93.3%). The maximum percentage of triggered aftershocks from the Higher ∆CFS (II) planes, is 78.3% (1σ: 61.5%–90.7%; 2σ: 34.8%–97.1%). Similarly, the percentage of aftershocks in positive ∆CFS considering the Reg + Eq Unstable (III) planes is 77.0% (1σ: 59.4%–89.3%; 2σ: 32.5%–96.6%). The percentage of aftershocks in positive ∆CFS for Reg Unstable (IV) planes is 68.0% (1σ: 49.8%–81.3%; 2σ: 26.2%–92.3%). The Strike Constrained (V) planes find 65.1% (1σ: 46.2%–80.3%; 2σ: 22.4%–93.5%) in positive ∆CFS. The minimum percentage of promoted aftershocks, from the Lower ∆CFS (I) planes, is 55.4% (1σ: 35.9%–73.2%; 2σ: 14.5%–89.4%).
As with the Italian earthquake sequences, the aftershocks in positive and negative ∆CFS are spatially similar between categories. The aftershocks in negative ∆CFS are intermixed with those in positive ∆CFS, and generally fall within several kms of the modeled rupture planes (Figure 4). The aftershocks in negative ∆CFS that extend further from the rupture planes highlight unmodelled cross-structures, or lie northwest of the Coso Geothermal area.
3.3 Comparison Between OOPs and Focal Mechanism Planes
The percentage of OOPs in positive ∆CFS is higher than the nominal results for focal mechanism planes, regardless of the nodal plane subset. For Umbria-Marche and L’Aquila, the percentage in positive ∆CFS for OOPs falls within the 2σ percentage range for all nodal plane categories. This is not the case for Ridgecrest, where OOPs lie outside of the 2σ percentage range for all but the planes in Higher ∆CFS (II) and the Reg + Eq Unstable (III) planes (Figure 3).
To compare OOP orientations with the observed focal mechanism planes, we compute the minimum angle between the pairs of normal and slip vectors for each aftershock (Figure 5). For Umbria-Marche, the OOPs and focal mechanism planes differ 24.4° on average between the closest plane normal vectors, and 35.1° between the closest slip vectors. For L’Aquila, the average difference between plane normal vectors is 26.0°, and the average between slip vectors is 37.7°. For Ridgecrest, the average difference between plane normals is 33.8°, while the average difference between slip vectors is 45.7°.
Angles between the closest OOP and nodal plane normal and slip vectors for all aftershocks of the three earthquake sequences.
3.4 Comparison Between Surface Structure and Nodal Plane Categories
The strikes from each plane category are displayed in the rose diagrams of Figure 6. Compared to the mapped surface structure, the Strike Constrained (V) planes expectedly reproduce the observed strike distributions best for all earthquake sequences. The Higher ∆CFS (II) planes produce flipped (in the case of Ridgecrest) or more scattered strikes (in the cases of L’Aquila and Umbria-Marche). The Reg Unstable (IV) planes better approximate the regional surface fault structure than the Reg + Eq Unstable (III) planes.
Rose diagrams of strikes for aftershock nodal plane categories of each earthquake sequence, as described in the main text. The diagram contours are individually labeled for the number of events in each 10° bin. Red lines mark peaks in mapped fault strikes (From Pizzi & Galadini, 2009 and faults from the USGS Qfaults database pictured in Figure 1). Thin circumferential black lines represent the range of strikes from mapped faults, where 180° differences in strike have no meaning for dip direction.
For individual aftershocks, the plane categories contain the same nodal planes less often than the similar spatial patterns of triggered vs. untriggered events might suggest, but agree with what we expect from the above comparison with surface structure (Table 2). Among the Strike Constrained (V) planes, the Reg Unstable (IV) planes, and the Reg + Eq Unstable (III) planes, 63.3% are common between categories for Umbria-Marche, 45.9% are common for L’Aquila, and 28.6% are common for Ridgecrest. Because the measures of instability are simply stepped versions of each other, it is most informative to next consider the Strike Constrained (V) planes compared with the planes picked based on measures of stress instability. For Umbria-Marche, the Strike Constrained (V) planes match best with the combined Reg + Eq Unstable (III) planes, picking the same planes 69.5% of the time, though the match with the Reg Unstable (IV) planes is only diminished to 66.1% (10 event difference). For L’Aquila and Ridgecrest, the Strike Constrained (V) planes match best with the Reg Unstable (IV) planes, with 56.6% and 45.6% matching, respectively. The matched planes diminish by ∼3% for both sequences when the Strike Constrained (V) planes are instead compared to the Reg + Eq Unstable (III) planes.
3.5 Contributions of Location and Orientation Uncertainties
Focal mechanism (i.e., hypocentral) location uncertainty introduces less variation to the percentage of aftershocks in positive ∆CFS than the plane orientation uncertainty. We separately test the uncertainty contributions for the Umbria-Marche sequence Strike Constrained (V) planes. The 1σ percentage range spans 7.8% with only location uncertainty, and 36.4% with only orientation uncertainty. Including the simultaneous contribution from both sources of uncertainty spans 37.0% for the 1σ percentage range. With uncertainties on strike, dip, and rake of 5° without location uncertainty, the 1σ percentage range diminishes to 8.8%, comparable to the range from the location uncertainty.
Because the strike, dip, and rake uncertainties are less well constrained and investigated than the location uncertainties for the focal mechanism data sets, we calculate results for the three earthquake sequences with different standard deviations for the orientation parameters. The location uncertainties are not changed, but the strike, dip, and rake standard deviations are set to equal values and tested at 5°, 10°, and 15° (Table S2 in Supporting Information S1). The span of the 1σ percentage range is >10% even with small (5°) standard deviations for strike, dip, and rake for each earthquake sequence (Table S2 in Supporting Information S1). For standard deviations of 15°, the 1 and 2σ percentage ranges are comparable to what we present here in the main text, with standard deviations stepped at 10°, 15°, and 30° for strike, dip, and rake, respectively.
3.6 Comparison With Synthetic Aftershock Sequences
For all aftershock sequences, the synthetic tests find a lower percentage of aftershocks in positive ∆CFS than for the observed sequences (Table S3 in Supporting Information S1). 46% are encouraged toward failure for Umbria-Marche, 51% for L’Aquila, and 44% for Ridgecrest. The results of the tests are sensitive to choice of aftershock spatial density, distribution, and extent of the modeled aftershock volume, so actual percentages of the synthetic tests must be considered cautiously, but the fact that the majority of points fall close to the rupture, as we observe in nature, lends confidence that the results can be generally compared with the actual sequences. An 8 km depth slice (Figure 4) of ∆CFS resolved on planes randomly sampled from each sequence's Strike Constrained (V) category reveals consistent spatial patterns of areas that tend to have more triggered or untriggered events, similar to what is predicted with the use of OOPs.
4 Discussion
For M > 5.5 events of the 1997 Umbria-Marche, 2009 L’Aquila (Italy) and 2019 Ridgecrest (California) earthquake sequences, we find accounting for aftershock receiver plane uncertainties allows us to statistically evaluate the OOPs' representativeness, and to support the significance of static ∆CFS calculations despite a large range of uncertainty (∼25%–30%) by noting the number of aftershocks associated with negative ∆CFS does not increase compared to random synthetic tests. With ∼50%–65% of nodal planes nominally triggered by static stress change, the assumption of OOPs overpredicts the percentage (+∼15%–30%) of aftershocks apparently promoted by coseismic static stress changes when compared to the observed nodal planes, and can lie outside of the 2σ percentage range of potentially triggered aftershocks, as demonstrated for the Ridgecrest earthquake sequence (Figure 3). We discuss the results considering the large range of potentially triggered events introduced by aftershock receiver plane uncertainties, our understanding (or lack thereof) of the structure related to aftershocks, and a retrospective look at choosing planes to evaluate earthquake hazard in the early stages of an aftershock sequence.
4.1 Impact of Receiver Plane Uncertainty on Static Triggering
Focal plane ambiguity, location uncertainties, and orientation uncertainties all increase the percentage range of statically triggered aftershocks, but we demonstrate that nodal plane ambiguity and orientation are of primary concern in evaluating the static stress triggering hypothesis. The outliers for aftershock percentages are the endmember categories of nodal planes in either lower or higher nominal ∆CFS. Neither endmember has a sound physical basis for representing the true fault at depth, but they provide an idea of the maximum spread of triggered aftershock percentages introduced by nodal plane ambiguity (∼20%–30%; Figure 3). For the categories of planes (I: Lower ∆CFS; II: Higher ∆CFS; III: Reg + Eq Unstable; IV: Reg Unstable; V: Strike Constrained; Table 2), the uncertainty introduced by location and orientation overtakes that of nodal plane ambiguity, with 1σ percentage ranges for triggered aftershocks spanning over 30% (Figure 3). The lower bound of triggered events is comparable to that of the random synthetic tests, and the upper bound approaches or surpasses the percentage of triggered events using OOPs. As demonstrated by tests for Umbria-Marche, the contribution of location uncertainty is relatively small compared to that of orientation uncertainty (7% vs. 36% for the 1σ percentage range for the separate evaluations), which approaches that from location uncertainty only if the orientation parameter standard deviations are smaller than current best estimates (5°; Table S2 in Supporting Information S1). The role of static stress triggering in earthquake sequences, therefore, requires careful consideration for both nodal plane ambiguity and orientation uncertainties, with secondary concern for location uncertainty.
While nodal plane ambiguity affects the percentage of triggered aftershocks, the spatial distribution of events encouraged or discouraged by coseismic static stress change is relatively insensitive to nodal plane choice (Figure 4; Figure S1 in Supporting Information S1). This is relevant for future applications of operational earthquake forecasting. The predictive power of Coulomb rate-and-state aftershock sequences improves when knowledge of existing structure, partially informed by focal mechanism planes, is incorporated in a range of possible orientations for receiver faults (Cattania et al., 2014, 2015, 2018; Mancini et al., 2020). Thus, focal plane ambiguity is unlikely to appreciably affect physically based aftershock forecasts that incorporate prior background seismicity and near real-time focal mechanisms as they become available. Even if the auxiliary nodal plane is chosen, the spatial distribution of the resulting forecast should remain similar.
4.2 The Structure of Aftershocks
The classic view of OOPs inadequately represents aftershock kinematic distribution, and hides potential structural variability critical to understanding deformation patterns in terms of subsidiary faults and fractures activated during the aftershock sequences. The normal and slip vector orientation differences between OOPs and nodal planes exceed ∼25° and ∼35°, respectively. The regional stress field uncertainty, in addition to nodal plane ambiguity and the natural roughness of faults complicates assessing the existence of OOPs, but the large number of events in our study suggests these angular discrepancies are significant. Our results support the idea that OOPs are less ubiquitous than often assumed, and do not adequately approximate aftershock rupture planes (e.g., Cattania et al., 2014; Hardebeck et al., 1998; McCloskey et al., 2003; Nostro et al., 2005; Segou & Parsons, 2020). Furthermore, despite the success of OOPs in predicting a higher percentage of triggered aftershocks, they hide the natural structural variability of observed faults and fracture orientations that result in spatially mixed areas of triggered and untriggered events (as shown by the spotted pattern in Figure 4). This, in effect, may also hide the potential of aftershocks occurring in stress shadow areas where OOPs predict no static triggering. Furthermore, if the regional stress field is poorly matched with the inherited structure, OOPs will be an even poorer representation of the spatial distribution and orientation of aftershock planes.
Constraining nodal planes by comparison with existing surface structure may provide a better, though incomplete, representation of the faults and fractures activated by aftershocks at depth. For Ridgecrest, the 80°/260° planes are nearly absent for all but the Strike Constrained plane category (Figure 6). Likewise, only the strike constraint reproduces observed structure well for the L’Aquila sequence, particularly the absence of north-striking faults (Figure 6; Pizzi & Galadini, 2009). Planes chosen on a basis of mechanical instability in the modeled stress field, while intuitive, do not reproduce these observations, introduce further ambiguity when both nodal planes are oriented for failure, and fall susceptible to our incomplete knowledge of the stress field. After all, the regional stress magnitude is not well known, nor are pre-existing or post-seismically induced small-scale heterogeneities (e.g., Cattania et al., 2018; Mancini et al., 2020; Vavrycuk, 2014). However, if the local structure is unknown, the best knowledge of the regional (and not the combined regional and earthquake) stress field will generally identify planes more consistent with mapped faults, at least in the cases presented here (Figure 6). The regional control is further supported by the observation that the distribution of nodal plane strikes does not change appreciably near or far from the rupture planes. We consider this finding with the caveat that faults inherited from previous tectonic stress regimes may not be well utilized by the current regional stress field if they are sub-optimally oriented for failure, but still in positive failure stress. Improvement on the strike constraint might be achieved by cluster analysis of aftershock hypocenters (e.g., Nandan et al., 2016) to choose planes that are common between neighboring aftershocks, and would consider the dip of existing planes. However, our choice of Strike Constrained planes seems a good approximation for the purpose of our study, particularly knowing that smaller, off-fault rupture planes remain likely and unpredictable.
A view of Strike Constrained aftershock planes separated by magnitude further validates choosing planes that align with mapped faults. Figure 7 shows contoured stereonets of slip vectors for the aftershocks of each earthquake sequence separated by magnitudes greater or less than 3. The scatter of smaller magnitude events becomes apparent, while the larger magnitudes more clearly align with what would be expected from mapped faults and regional kinematics. For Ridgecrest, the larger events primarily slip subhorizontally, consistent with the expected strike-slip environment. For Umbria-Marche and L’Aquila, the slip vectors concentrate on ∼40°–60° dipping planes, usually dipping to the southwest, though fewer northeast-dipping, likely conjugate structures are highlighted as well. The larger scatter in slip vectors for the Umbria-Marche M > 3 events, deviating from pure normal faulting on northwest/southeast dipping planes, is primarily from events shallower than ∼4 km. This may indicate alteration of the stress field influenced by the hypothesized deep, high-pressure CO2 source region investigated in Miller et al. (2004), causing faults to slip at shallower rake angles. It might also indicate stress heterogeneity caused by geometrical complexities of the segmented normal faults (Ripepe et al., 2000). While the larger magnitude events are generally more consistent with observed structure, the percentage of triggered events for M > 3 is not appreciably different (<7% change) when compared to the triggering percentages for aftershocks of M < 3.
Slip vector contour plots for the Strike Constrained nodal planes separated by M ≥ 3 and M < 3. The slip vectors are presented in equal-area projections with 2σ Kamb contouring. The number of slip vectors in each color contour is labeled to the right of each plot.
4.3 Assessing Static Triggering Amidst Data and Model Deficiencies
The scatter of slip vector orientations for smaller events could indicate stress and/or structural heterogeneity in the crust. Events of M < 3 are expected to be on the order of 102 m or less in source dimension, with slip on the order of 10−2 m or less (e.g., Hanks & Bakun, 2002). We hypothesize that the smaller magnitude events tend to occur not only on the larger active structures consistent with the current tectonic regime, but also on subsidiary or inherited structures susceptible to smaller-scale crustal stress heterogeneities. Faults and fractures within brittle shear zones are known to exhibit systematic orientation variations relative to the primary slip surface (e.g., Reidel, 1929). This is also consistent with observations of Brunsvik et al. (2021), which notes greater angular mismatch between the Paganica fault of the L’Aquila mainshock and smaller magnitude aftershocks near the rupture plane. Other processes in the vicinity of the mainshock rupture planes, such as dynamic rupture propagation and fluid flow, are likely to alter the stress state for those aftershocks. However, we also note that these smaller events may be more susceptible to high orientation errors. Even if the strike constraint is poor for small magnitude events, it is arguably an important physical consideration for the larger magnitude events that pose a greater hazard.
We caution, however, that the density of events inconsistent with static triggering near the mainshock rupture planes could signify deficiencies in our models of static stress change and/or the importance of other triggering mechanisms. Uncertainties associated with the modeled rupture have measurable effect on evaluation of the static stress triggering hypothesis (e.g., Cattania et al., 2014; Steacy et al., 2004; Wang et al., 2014). Near the mainshock fault, the stress field is particularly sensitive to geometric complexity, including bends and linkages (e.g., Maerten et al., 2002), as well as to the discretization of modeled slip (e.g., Meade, 2007). The spatial distribution of unpromoted aftershocks may also indicate other fault zone processes that modify the near-field stresses or diminish the frictional resistance of faults to encourage aftershocks toward failure. For example, afterslip has been inferred for both L’Aquila and Ridgecrest, which would modify the static stress field (Cheloni et al., 2014; Qiu et al., 2020). Similarly, poroelastic response to high-pressure fluids and fluid flow has been proposed as instrumental in promoting aftershocks following the Umbria-Marche and L’Aquila earthquake sequences (Miller et al., 2004; Terakawa et al., 2010). Secondary triggering by preceding aftershocks is also well documented to contribute toward the improvement of physically based aftershock forecasts (e.g., Mancini et al., 2020). This mechanism potentially alters both the near- and far-field stresses. Our analysis may demonstrate a large uncertainty for the mainshock-static stress triggering hypothesis, but does not negate the role of these other processes in also promoting aftershocks toward failure.
Potential model deficiencies in the near-field of the rupture planes are further highlighted by separating aftershocks with distance from the fault; however, uncertainties from receiver plane orientation remain high regardless of the distance from the source. Excluding events closer than 5 km, consistent with other stress-triggering studies (e.g., Hardebeck et al., 1998; King & Cocco, 2001; McCloskey et al., 2003; Steacy et al., 2004; Wang et al., 2014), markedly increases the nominal percentage of events promoted by static stress change. Of the 19 Strike Constrained focal planes available beyond 5 km of the modeled rupture planes for the Umbria-Marche sequence, 63% are nominally promoted, while only 50% (of 300) of the aftershocks within 5 km are promoted toward failure. For L’Aquila, the percentage of promoted events jumps from 52% (of 2,045) to 87% (of 1,370) for aftershocks beyond 5 km. Similarly, for Ridgecrest, the triggering percentage increases from 60% (of 2,355) to 87% (of 534) beyond 5 km. This distance effect on triggering percentage was unclear or non-existent for the synthetic sequences, suggesting a true static control on the occurrence of aftershocks that dominates with distance from the planes (Table S3 in Supporting Information S1). This again may demonstrate deficiencies in the modeled near-field stresses from processes such as afterslip, fluid-related effects, or missing details in the models of static stress change. Notably, the 1σ percentage range of triggered aftershocks is large regardless of the distance to the coseismic rupture planes, indicating these uncertainties must be accounted for in the near- and far-field. Uncertainty near mainshock rupture planes is a long-standing problem in evaluating static stress triggering. We show that this uncertainty stems not only from unmodelled or mismodeled fault plane and near-source processes, but also from receiver plane uncertainties.
Large uncertainty for static stress triggering demonstrated here from receiver plane uncertainties alone makes the hypothesis difficult to discount. And, while the percentage range of aftershocks triggered by mainshock static stress change is large, there is growing evidence that over half of the aftershocks associated with any given sequence are encouraged toward failure, with estimates more commonly near two-thirds for various aftershock sequences. Umbria-Marche falls toward the lower end, with ∼50% of the Strike Constrained planes in positive failure for the given plane locations and orientations. It is worth noting that network density at the southernmost sector of the fault system had incomplete coverage, limiting the available aftershock solutions (see event concentration in Figure 4). In contrast, ∼65% of the L’Aquila and Ridgecrest Strike Constrained nodal planes are promoted toward failure. Other studies including static stress triggering resolved on aftershock focal mechanism planes find similar results with focal mechanism solutions (∼60%–70% triggered), including the 2003 Bam, Iran earthquake (Asayesh et al., 2020), the 1992 Landers, California earthquake (Steacy et al., 2004), and the 1994 Northridge earthquake (Hardebeck et al., 1998). With an energetics-based failure stress approach that circumvents the issue of nodal plane ambiguity, Terakawa et al. (2020) similarly find ∼68% of aftershocks consistent with static triggering for the Landers event. This list is not exhaustive, but indicates a strong correspondence with more than half of aftershocks promoted by static stress change from the largest mainshocks.
5 Conclusions
For the 1997 Umbria-Marche, 2009 L’Aquila (Italy), and 2019 Ridgecrest (California) earthquake sequences, the largest mainshocks encourage ∼50%–65% of aftershocks toward failure. The uncertainties associated with nodal plane orientation, nodal plane ambiguity, and, to second order, hypocentral location expand the percentage of potentially triggered aftershocks over 30% for the 1σ range. Surprisingly, accounting for receiver plane uncertainties does not increase the number of aftershocks associated with negative Coulomb stress changes relative to synthetic tests. The 2σ percentage ranges for L’Aquila and Ridgecrest show nearly any of the observed aftershocks could have been triggered by static stress transfer. The nominal percentage of events also markedly increases in the far-field, at distances greater than 5 km from the coseismic rupture planes. Typically, untriggered aftershocks in the near-field are attributed to mis- or un-modeled geometrical complexities and fault zone processes. We demonstrate that receiver plane uncertainty is equally, or even potentially more, important to consider.
The structure activated by aftershocks highlights additional need for understanding regional faults and fracture networks, and their variability at the sub-kilometer scale. The classic view of OOPs for failure in the combined regional and earthquake perturbed stress field overestimates the role of static stress triggering, with planes significantly differing in orientation from observed nodal planes. Instead, the aftershocks align best in strike with pre-existing mapped surface faults, though the plane orientations and slip directions of smaller magnitude events (M < 3) are more heterogeneous. A better understanding of focal mechanism orientation errors and the structure of smaller-scale faults could strengthen the assessment of the role of static stress triggering. Nonetheless, the correspondence of our nominal results with past analyses of the static stress triggering hypothesis indicates over half of the aftershocks are promoted by static stress change, with more common estimates nearing two-thirds, regardless of magnitude.
Acknowledgments
This work was supported by the NSF grant 1723045 to the University of Arizona, and scholarships awarded through the University of Arizona Department of Geosciences (the Sulzer and Sumner Scholarships). Massimo Cocco participated to this work as Principal Investigator of the European Research Council (ERC) project FEAR (grant 856559) under the European Community's Horizon 2020 Framework Programme. Many figures were prepared with the GMT software (Wessel et al., 2013) or the Python Matplotlib package (Hunter, 2007). All focal mechanism solutions, coseismic slip distributions, and software used are available as designated in the original publications, cited in the methods section. The authors also thank two anonymous reviewers, the associate editor, and editor Rachel Abercrombie, whose comments lead to valuable additions. We also thank Katherine Guns, Eric Kiser, and Susan Beck for helpful discussions and guidance.
Open Research
Data Availability Statement
The aftershock information and modeled coseismic stress changes for this analysis are available on GitHub (https://zenodo.org/record/6289582). Interactive 3D plots of the results are available on Observable: https://observablehq.com/@cehanagan/aftershock_stress.
