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Examples of two static stress fields considered in this analysis and corresponding ROC curves for one slip distribution (Cohee & Beroza, 1994) from the 1992 M w = 7.3 Landers earthquake in California. (a-c) Map view of ΔCFS(σ, 0.4) values within 100 km of the fault at 2.5 km, 7.5 km, and 22.5 km depth, respectively. Black squares represent grid cells in which one or more aftershocks occurred within 1 year of the mainshock. Thick yellow and black line represents extent of the mainshock rupture at each depth. Scale bar is shown in Figure 1a as a thick black line. (d) ROC curve for this particular slip distribution (Cohee & Beroza, 1994) and ΔCFS(σ, 0.4), including all grid cells and all aftershocks within a year after the mainshock. Black dotted line is a 1:1 line for reference. (e-h) Analogous to Figures 1a-1d for a different static stress field, Δτ max (σ).
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Aftershocks may be triggered by the stresses generated by preceding mainshocks. The temporal frequency and maximum size of aftershocks are well described by the empirical Omori and Bath laws, but spatial patterns are more difficult to forecast. Coulomb failure stress is perhaps the most common criterion invoked to explain spatial distributions of a...
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Context 1
... AUC analysis, the stress metrics can be ranked in order of their ability to correctly classify whether or not an aftershock occurred in a grid cell. Consider the ROC curves for one of the 1994 M w = 7.4 Landers earthquake slip distributions (Figure 1) (Cohee & Beroza, 1994): for maximum shear stress (Figure 1h), the ROC curve is above the 1:1 line with AUC = 0.87, suggesting that this stress metric can be used to discriminate grid cells with and without aftershocks in the volume around the Landers' mainshock more accurately than a random classifier. By contrast, the corresponding ROC curve for classic Coulomb failure stress (AUC = 0.61; Figure 1d) suggests that using this scalar mechanical metric to classify grid cells is little better than random assignment. ...
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... AUC analysis, the stress metrics can be ranked in order of their ability to correctly classify whether or not an aftershock occurred in a grid cell. Consider the ROC curves for one of the 1994 M w = 7.4 Landers earthquake slip distributions (Figure 1) (Cohee & Beroza, 1994): for maximum shear stress (Figure 1h), the ROC curve is above the 1:1 line with AUC = 0.87, suggesting that this stress metric can be used to discriminate grid cells with and without aftershocks in the volume around the Landers' mainshock more accurately than a random classifier. By contrast, the corresponding ROC curve for classic Coulomb failure stress (AUC = 0.61; Figure 1d) suggests that using this scalar mechanical metric to classify grid cells is little better than random assignment. ...
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... the ROC curves for one of the 1994 M w = 7.4 Landers earthquake slip distributions (Figure 1) (Cohee & Beroza, 1994): for maximum shear stress (Figure 1h), the ROC curve is above the 1:1 line with AUC = 0.87, suggesting that this stress metric can be used to discriminate grid cells with and without aftershocks in the volume around the Landers' mainshock more accurately than a random classifier. By contrast, the corresponding ROC curve for classic Coulomb failure stress (AUC = 0.61; Figure 1d) suggests that using this scalar mechanical metric to classify grid cells is little better than random assignment. ...
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... mechanical quantity. The 21 top-ranked Category I stress metrics are everywhere positive ( Figure A1), and thus, it is possible that distance is the most important control on aftershock distributions, and all the other Category I stress metrics simply correlate well with distance. ...
Context 5
... stress metrics are corrected for distance as 1/r 2 , consistent with the decay expected for a finite source (Okada, 1992). (In the far-field, stresses would be expected to decrease as 1/r 3 from a point source (Okada, 1992), but because the vast majority aftershocks take place near (<~50 km from) the mainshock (e.g., Figure 1), this correction is likely not be appropriate here.) Once corrected for distance as 1/r 2 , the only metrics that consistently (Ψ > 0.70) perform significantly better (α = 0.005) than random classifiers consistently are the magnitudes of the second and third invariants of the full and deviatoric stress tensor. ...
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Citations
... Numerous studies have shown a strong correlation between aftershocks spatial pattern and changes in static Coulomb Failure Stress (CFS) resulting from mainshocks (e.g., Asayesh et al., 2018Asayesh et al., , 2019Asayesh, Zarei, & Zafarani, 2020;Harris & Simpson, 1992;Jamalreyhani et al., 2020;King et al., 1994;Toda et al., 1998). Recently, some studies using receiver operating characteristic (ROC) analysis have shown that several alternative scalar metrics are better predictors of aftershock locations than ΔCFS (DeVries et al., 2018;Meade et al., 2017;Mignan & Broccardo, 2019). Later, Sharma et al. (2020), by using available slip models from the SRCMOD database (Mai & Thingbaijam, 2014) and repeating the analysis with more appropriate stress calculations, showed that ΔCFS resolved on optimally oriented planes (OOP) or calculated for variable mechanism (VM) significantly improved the ΔCFS results for receiver mechanisms identical to the mainshock mechanism (on master fault orientation, MAS). ...
The Epidemic Type Aftershock Sequence (ETAS) model is the most widely used and powerful statistical model for aftershock forecasting. While the distribution of aftershocks around the mainshock is anisotropic, the spatial probability density function of the ETAS model is commonly assumed to be isotropic due to insufficient information. In addition, its parameter estimation can be highly biased due to catalog incompleteness after the mainshock. Thus, we extended the recently developed 2D temporal ETASI, which accounts for short‐term incompleteness, to 2D and 3D spatiotemporal ETASI, considering additional spatial occurrence probabilities in the framework of ETAS and ETASI to improve aftershock forecasting. We replaced the isotropic spatial kernel with anisotropic kernels estimated by a spatial probability map of stress scalars, including Coulomb stress changes on master fault orientation (MAS), Coulomb stress changes on variable mechanisms (VM), maximum shear (MS), and von Mises stress (VMS), and the nearest distance to the ruptured fault of the mainshock (R). The fit to six prominent mainshock‐aftershock sequences in California demonstrates that the ETASI model outperforms the standard ETAS model. Furthermore, positive information gains indicate that using stress calculations as additional input information can improve the parameter fit. This improvement is weaker in 3D, which is likely related to greater positional uncertainty in the depth domain. However, incorporating the probability map calculated as a function of the nearest distance to the mainshock rupture leads to the best performance in all model variants.
... To conclude, such investigations of fault slip mechanics and aftershocks are benefited of a detailed fault structure constructed in this study, especially at fault step-overs, joints, and bends. This implies that the existence of unexplained aftershocks by positive ΔCFS or unexpected slip terminations of numerous previous earthquakes (e.g., Meade et al., 2017;Segou and Parsons, 2014;Storti et al., 2007) may be due to, if not entirely, the lack of detailed fault structural information which is worth re-investigating. With the advances of geodetic and geophysical observations, it is suggested that such analysis should be conducted routinely for future large events and to uncover more properties of the Earth's shallow crust. ...
... The distribution of aftershocks is not uniform but associated with the inhomogeneous stress changes induced by the mainshock (Reasenberg and Simpson, 1992;Deng and Sykes, 1996;Meade et al., 2017). In particular, the spatial distribution of aftershocks generally correlates with positive Coulomb stress changes (King et al., 1994;Asayesh et al., 2019Asayesh et al., , 2020b. ...
... This group has also developed community-vetted testing protocols and metrics. In addition to the CSEP testing metrics, the receiver operating characteristic (ROC) curve (Hanley and McNeil, 1982) is widely applied to assess the performance of aftershock forecasts based on primary physics-based models, including the Coulomb forecast ( CFS) model, neural network predictions, and the distance-slip model (Meade et al., 2017;DeVries et al., 2018;Mignan and Broccardo, 2019;Sharma et al., 2020;Asayesh et al., 2022). The ROC is based on a binary classification of test events referred to as observed earthquakes. ...
... Recently, the ability of CFS models to forecast aftershock locations has been questioned by using the ROC curve in comparison to various scalar metrics, distance-slip and deep neural network (DNN) models (Meade et al., 2017;DeVries et al., 2018;Mignan and Broccardo, 2019;Sharma et al., 2020;Asayesh et al., 2022). These studies showed that several alternative scalar stress metrics, which do not need any specification of the receiver mechanism, and a simple distance-slip model as well as DNN are better predictors of aftershock locations than CFS for fixed receiver orientation. ...
Aftershock forecast models are usually provided on a uniform spatial grid, and the receiver operating characteristic (ROC) curve is often employed for evaluation, drawing a binary comparison of earthquake occurrences or non-occurrence for each grid cell. However, synthetic tests show flaws in using the ROC for aftershock forecast ranking. We suggest a twofold improvement in the testing strategy. First, we propose to replace ROC with the Matthews correlation coefficient (MCC) and the F1 curve. We also suggest using a multi-resolution test grid adapted to the earthquake density. We conduct a synthetic experiment where we analyse aftershock distributions stemming from a Coulomb failure (ΔCFS) model, including stress activation and shadow regions. Using these aftershock distributions, we test the true ΔCFS model as well as a simple distance-based forecast (R), only predicting activation. The standard test cannot clearly distinguish between both forecasts, particularly in the case of some outliers. However, using both MCC-F1 instead of ROC curves and a simple radial multi-resolution grid improves the test capabilities significantly. The novel findings of this study suggest that we should have at least 8 % and 5 % cells with observed earthquakes to differentiate between a near-perfect forecast model and an informationless forecast using ROC and MCC-F1, respectively. While we cannot change the observed data, we can adjust the spatial grid using a data-driven approach to reduce the disparity between the number of earthquakes and the total number of cells. Using the recently introduced Quadtree approach to generate multi-resolution grids, we test real aftershock forecast models for Chi-Chi and Landers aftershocks following the suggested guideline. Despite the improved tests, we find that the simple R model still outperforms the ΔCFS model in both cases, indicating that the latter should not be applied without further model adjustments.
... However, the spatial distribution of aftershocks around the mainshock is generally not isotropic. Many studies show that anisotropic spatial distribution of aftershocks correlates well with coseismic stress changes due to mainshock (e.g., Asayesh, Zarei, et al., 2020;Hainzl et al., 2010;King et al., 1994;Meade et al., 2017;Sharma et al., 2020). The assumption of isotropic aftershock distributions in the ETAS model can significantly bias the estimation of the model parameters, in particular, the α value (Hainzl et al., 2008). ...
Current earthquake catalogs provide high‐precision depth values that contain valuable information. Thus, we extend the spatiotemporal Epidemic Type Aftershock Sequence model to 3D by considering hypocentral instead of epicentral distances. To explore the most appropriate parametric form of the spatial kernel, we first examine different triggering functions for the depth difference of the aftershock‐mainshock pairs identified by the Nearest‐Neighbor method, showing that a magnitude‐dependent power‐law kernel fits best the earthquakes data in Southern California. Therefore, we incorporate the corresponding kernel into the 3D‐ETAS model with space‐dependent background activity, where we additionally allow for depth‐dependent aftershock productivity. The application to Southern California shows that the model fits the data well. We also find that the aftershock productivity strongly depends on depth, similar to the seismic moment released by the mainshocks, which may be related to depth‐dependent seismic coupling.
... The Coulomb Failure Stress (CFS) is a commonly-used scalar that quantifies the stress state and then evaluates rock failure, earthquake nucleation and fault reactivation. The value of CFS is dependent on the initial stress state and may involve some uncertainties (Meade et al., 2017). In contrast, the magnitude of ΔCFS does not depend on the initial conditions; instead, it is determined by the variation of the stress tensor after normalizing it with respect to the assumed parameters and the perturbations affecting the system (De Simone et al., 2017; Wang et al., 2014). ...
The Coulomb failure stress (CFS) criterion based on the Mohr-Coulomb friction theory is commonly used to explain physical mechanisms governing injection-induced seismicity. While the CFS criterion can evaluate the onset of fault reactivation caused by fluid perturbations, it cannot tell the kinematic process of fault failure (seismic or aseismic slip). An alternative model for simulating time-dependent earthquake cycles is the rate-and-state friction model, which provides evolving friction depending on slip rate and slip history. To explore the dynamics and stability of fault slip associated with fluid injection, we extend a spring-slider system with a rate-and-state dependent friction law by incorporating the evolving fluid pressure and poroelastic stress. Simulations of continuous constant fluid mass injection rate scenarios using the developed model suggest that injection-induced fault slip behavior is controlled by fault orientation, diffusivity and poroelasticity, and injection rate. Compared to the CFS criterion, our model can offer a temporal component to fault friction and provide new insights on slip, slip rate and trajectory in phase space. We also investigate the relation between cumulative slip displacement versus cumulative injection volume, as well as the event recurrence interval. Our proposed approach has the potential application for evaluating the reactivation of pre-existing faults embedded in the reservoir associated with fluid pressure operations in the field practice.
INTRODUCTION
It has been well established that fluid injection into subsurface reservoirs can induce earthquakes (Healy et al., 1968; Raleigh et al., 1976). One of the most frequently discussed scientific issues is the physical mechanism that causes earthquakes to nucleate in response to fluid injection. The Coulomb failure stress criterion based on the Mohr-Coulomb friction theory is commonly used to explain the mechanism of induced seismicity (Ge & Saar, 2022). The Coulomb failure stress (CFS) and the change in Coulomb failure stress (ΔCFS) can be expressed as
(Equation)
(Equation)
where μ is the friction coefficient that is assumed to be constant above, τ, σn and P refer to shear stress, normal stress (positive for compression), and pore pressure acting on the fault, respectively. Δτ, Δσn and ΔP denote the changes of shear stress, normal stress and pore pressure, respectively.
... In this study, we analyze the GR and OU parameters as a function of the induced stress change and rupture distance for the combined data set of 127 mainshock-aftershock sequences. In doing so, we not only adhere to calculations of Coulomb Failure Stress changes (∆CFS) with alternative definitions of receiver mechanisms, but we also compute receiver-independent stress metrics, recently shown to be superior in forecasting spatial aftershock distributions (Meade et al. 2017;Sharma et al. 2020). Our results show systematic stress-and distance-dependent variations of the GR and OU parameters that can be useful for aftershock hazard estimations. ...
... Specifically, we use the PSGRN + PSCMP code by (Wang et al. 2006) to calculate the stress change tensor on a regularly gridded volume with a grid spacing of 5 km in all three dimensions. Using the calculated stress change tensor, we compute the Coulomb failure stress (∆CFS) at each grid point based on different assumptions for the receiver mechanism (MAS, OOP, VM) and two alternative simple stress scalars (MS, VMS), which are good predictors of the spatial aftershock distribution (Meade et al. 2017;Sharma et al. 2020). In particular, we determine the following five stress metrics: ...
The Gutenberg-Richter (GR) and the Omori-Utsu (OU) law describe the earthquakes’ energy release and temporal clustering and are thus of great importance for seismic hazard assessment. Motivated by experimental results, which indicate stress-dependent parameters, we consider a combined global dataset of 127 mainshock-aftershock sequences and perform a systematic study of the relationship between mainshock-induced stress changes and associated seismicity patterns. For this purpose, we calculate space-dependent Coulomb Stress (ΔCFS) and alternative receiver-independent stress metrics in the surrounding of the mainshocks. Our results indicate a clear positive correlation between the GR b-value and the induced stress, contrasting expectations from laboratory experiments and suggesting a crucial role of structural heterogeneity and strength variations. Furthermore, we demonstrate that the aftershock productivity increases nonlinearly with stress, while the OU parameters c and p systematically decrease for increasing stress changes. Our partly unexpected findings can have an important impact on future estimations of the aftershock hazard.
... The distribution of aftershocks is not uniform but associated with the inhomogeneous stress changes induced by the mainshock (Reasenberg and Simpson, 1992;Deng and Sykes, 1996;Meade et al., 2017). In particular, the spatial distribution of 25 aftershocks generally correlates with positive Coulomb stress changes (King et al., 1994;Asayesh et al., 2019Asayesh et al., , 2020b. ...
... This group has also developed community-vetted testing protocols and metrics. In addition to the CSEP testing metrics, the 40 Receiver Operating Characteristic (ROC) curve is widely applied to assess the performance of aftershock forecasts based on primary physics-based models, including the Coulomb forecast (∆CF S) model, neural network predictions, and the distanceslip model (Meade et al., 2017;DeVries et al., 2018;Mignan and Broccardo, 2019;Sharma et al., 2020;Asayesh et al., 2022). ...
... Recently, the ability of ∆CF S models to forecast aftershock locations has been questioned by using the ROC curve in comparison to various scalar metrics, distance-slip and deep neural network (DNN) models (Meade et al., 2017;DeVries et al., 2018;Mignan and Broccardo, 2019;Sharma et al., 2020;Asayesh et al., 2022). These studies showed that several alternative 60 scalar stress metrics which do not need any specification of the receiver mechanism, and a simply distance-slip model as well as DNN are better predictors of aftershock locations than ∆CF S for fixed receiver orientation. ...
Aftershock forecast models are usually provided on a uniform spatial grid, and the receiver operating characteristic (ROC) curve is often employed for evaluation, drawing a binary comparison of earthquake occurrences or non-occurrence for each grid cell. However, synthetic tests show flaws in using ROC for aftershock forecast ranking. We suggest a twofold improvement in the testing strategy. First, we propose to replace ROC with the Matthews correlation coefficient (MCC) and the F1 curve. We also suggest using a multi-resolution test grid adapted to the earthquake density. We conduct a synthetic experiment where we analyze aftershock distributions stemming from a Coulomb Failure (∆CFS) model, including stress activation and shadow regions. Using these aftershock distributions, we test the true ∆CFS model as well as a simple distance-based forecast (R), only predicting activation. The standard test cannot clearly distinguish between both forecasts, particularly in the case of some outliers. However, using both MCC-F1 instead of ROC curves and a simple radial multi-resolution grid improves the test capabilities significantly. Our findings suggest that to conduct meaningful tests, we should have at least 8 % and 5 % cells with observed earthquakes to differentiate between a near-perfect forecast model and an informationless forecast using ROC and MCC-F1, respectively. While we cannot change the observed data, we can adjust the spatial grid using a data-driven approach to reduce the disparity between the number of earthquakes and the total number of cells. Using the recently introduced Quadtree approach to generate multi-resolution grids, we test real aftershock forecast models for Chi-Chi and Landers aftershocks following the suggested guideline. Despite the improved tests, we find that the simple R model still outperforms the ∆CFS model in both cases, indicating that the latter should not be applied without further model adjustments.
... where σ s is the shear stress, μ is the average shear modulus, P pore is the change of pore pressure and σ n is the normal stress, has been applied to correlate its variations with changes in aftershocks productivity [313][314][315][316]. Recently, some concerns have been advanced about the real effectiveness of this parameter to chart the spatial distributions of stress gradients in the crust, while others have been proposed [317,318]. A difference between static and dynamic Coulomb stress is conventionally done: when loading is slow, so that its increasing/decreasing rate is negligible with respect to the compared time interval, then the static Coulomb stress is at work, on the contrary, if loading occurs suddenly (i.e., fluid injection, coseismic slip), then the dynamic Coulomb failure stress plays a relevant role. ...
The processes occurring on the Earth are controlled by several gradients. The surface of the Planet is featured by complex geological patterns produced by both endogenous and exogenous phenomena. The lack of direct investigations still makes Earth interior poorly understood and prevents complete clarification of the mechanisms ruling geodynamics and tectonics. Nowadays, slab-pull is considered the force with the greatest impact on plate motions, but also ridge-push, trench suction and physico-chemical heterogeneities are thought to play an important role. However, several counterarguments suggest that these mechanisms are insufficient to explain plate tectonics. While large part of the scientific community agreed that either bottom-up or top-down driven mantle convection is the cause of lithospheric displacements, geodetic observations and geodynamic models also support an astronomical contribution to plate motions. Moreover, several evidences indicate that tectonic plates follow a mainstream and how the lithosphere has a roughly westerly drift with respect to the asthenospheric mantle. An even more wide-open debate rises for the occurrence of earthquakes, which should be framed within the different tectonic setting, which affects the spatial and temporal properties of seismicity. In extensional regions, the dominant source of energy is given by gravitational potential, whereas in strike-slip faults and thrusts, earthquakes mainly dissipate elastic potential energy indeed. In the present article, a review is given of the most significant results of the last years in the field of geodynamics and earthquake geology following the common thread of gradients, which ultimately shape our planet.
... Recently, some studies using ROC analysis have further questioned the ability of CFS values in forecasting aftershock locations. Performing a thorough analysis of various scalar metrics, Meade et al. (2017) showed that a number of alternative scalar metrics, which do not need any specification of the receiver mechanism, are better predictors of aftershock locations than CFS for fixed receiver orientation. Those superior stress scalars comprise the maximum shear, the von-Mises yield criterion (a scaled version of the second invariant of the deviatoric stress-change tensor), and the sum of the absolute values of the independent components of the stress-change tensor. ...
... The most common ways for assessing and comparing binary classifier performance are the receiver operating characteristic (ROC) curve (Kleinbaum & Klein 1994) and the precision-recall (PR) curve (Raghavan et al. 1989). In previous analyses of aftershocks spatial forecasts, the ROC curves were used to evaluate the forecasting capability of different methods such as primary physicsbased models including CFS, neural network predictions, and the R-model ( Despite the many advantages of the ROC curves over a statistic like the mis-classification rate (Meade et al. 2017), some studies show that the ROC analysis on imbalanced data can be misleading (Davis et al. 2005;Davis & Goadrich 2006;Jeni et al. 2013;Saito & Rehmsmeier 2015). In particular, Parsons (2020) discussed that the previous ROC tests for spatial aftershock forecasts suffer from data imbalance because most areas lack any event (many more negative than positive cases), inhibiting the resolving power of the ROC analysis. ...
... Their occurrence is mainly related to the static stress changes of the main shock, although other effects such as dynamic or post-seismic processes and alterations of fault strengths can also play an important role. Recent analyses of binary forecasts of the aftershock area revealed that the classical approach of calculating CFS is not the best and outperformed by receiverindependent stress scalars (Meade et al. 2017;Sharma et al. 2020). However, these results were questioned because of the applied ROC analysis for testing, which might suffer from data imbalance (Parsons 2020). ...
On 12 November 2017, an earthquake with a moment magnitude of 7.3 struck the west of Iran near the Iraq border. This event was followed about 9 and 12 months later by two large aftershocks of magnitude 5.9 and 6.3, which together triggered intensive seismic activity known as the 2017-2019 Kermanshah sequence. In this study, we analyze this sequence regarding the potential to forecast the spatial aftershock distribution based on information about the mainshock and its largest aftershocks. Recent studies showed that classical Coulomb failure stress (CFS) maps are outperformed by alternative scalar stress quantities, as well as a distance-slip probabilistic model (R) and deep neural networks (DNN). In particular, the R-model performed best. However, these test results were based on the receiver operating characteristic (ROC) metric, which is not well suited for imbalanced data sets such as aftershock distributions. Furthermore, the previous analyses also ignored the potential impact of large secondary earthquakes. For the complex Kermanshah sequence, we applied the same forecast models but used the more appropriate MCC-F1 metric for testing. Similar to previous studies, we also observe that the receiver independent stress scalars yield better forecasts than the classical CFS values relying on the specification of receiver mechanisms. However, detailed analysis based on the MCC-F1 metric revealed that the performance depends on the grid size, magnitude cutoff, and test period. Increasing the magnitude cutoff and decreasing the grid size and period reduce the performance of all methods. Finally, we found that the performance of the best methods improves when the source information of large aftershocks is additionally considered, with stress-based models outperforming the R model. Our results highlight the importance of accounting for secondary stress changes in improving earthquake forecasts.
... Their follow-up study assessed their injection event determination model against several machine learning approaches, showing that the ROC curves for their model are good or better than "black box" approaches (i.e., including physics often helps with event classification). Sliding thresholds are used in earth science studies, too, for example, when Meade et al. (2017) swept model event settings to determine which stress metrics are most effective at predicting aftershocks following major earthquakes. ...
We apply idealized scatterplot distributions to the sliding threshold of observation for numeric evaluation (STONE) curve, a new model assessment metric, to examine the relationship between the STONE curve and the underlying point‐spread distribution. The STONE curve is based on the relative operating characteristic (ROC) curve but is developed to work with a continuous‐valued set of observations, sweeping both the observed and modeled event identification threshold simultaneously. This is particularly useful for model predictions of time series data as is the case for much of terrestrial weather and space weather. The identical sweep of both the model and observational thresholds results in changes to both the modeled and observed event states as the quadrant boundaries shift. The changes in a data‐model pair's event status result in nonmonotonic features to appear in the STONE curve when compared to an ROC curve for the same observational and model data sets. Such features reveal characteristics in the underlying distributions of the data and model values. Many idealized data sets were created with known distributions, connecting certain scatterplot features to distinct STONE curve signatures. A comprehensive suite of feature‐signature combinations is presented, including their relationship to several other metrics. It is shown that nonmonotonic features appear if a local spread is more than 0.2 of the full domain or if a local bias is more than half of the local spread. The example of real‐time plasma sheet electron modeling is used to show the usefulness of this technique, especially in combination with other metrics.
