Evaluation of the b Maps on the Faults of the Major (M > 7) South California Earthquakes

We use the Godano et al. (2022, https://doi.org/10.1029/2021ea002205) method for evaluating the b maps of the faults associated with the largest earthquakes M ≥ 7.0 that occurred in California. The method allows an independent evaluation of the b parameter, avoiding the overlap of the cells and the omission of some earthquakes, while keeping all the available information in the catalog. We analyzed four large earthquakes: Landers, Hector Mine, Baja California, and Searles Valley. The maps obtained confirm that the b value can be considered as a strain meter and allow us to elucidate the presence of barriers, such as obstacles to the propagation of the fracture, on the fault of the analyzed earthquakes. A further estimated parameter is the time window during which aftershocks occur in the cell, Δt. This quantity is very useful for a better definition of the aftershock generation mechanism. It reveals where the stress is released in a short time interval and how the complexity of the faulting process controls the occurrence of aftershocks on the fault, and also the duration of the entire sequence.


Introduction
The evaluation of the b value of the Gutenberg and Richter (GR) distribution (Gutenberg & Richter, 1944) represents a crucial point in the estimation of earthquake hazard.
The b parameter has been extensively studied for earthquake catalogs and laboratory experiments.Its value has been inversely correlated with the stress state (Amitrano, 2003;Gulia & Wiemer, 2010;Scholz, 1968;Wyss, 1973) and directly correlated with the thermal gradient (Warren & Latham, 1970;Wiemer et al., 1998) as well as the material heterogeneity (Mogi, 1962).b value have been investigated both in time and space.
The time variation of the b value has been used mainly to characterize aftershock sequences and distinguish between aftershocks and foreshocks.Generally, b time variations have been investigated to elucidate the differences between foreshocks and background seismic activity (Ogata et al., 1996;Papadopoulos et al., 2018;Papazachos, 1973Papazachos, , 1975;;Suyehiro & Sekiya, 1972).Conversely, Gulia et al. (2018), by stacking many earthquake sequences, found that the b value increases immediately after the occurrence of a mainshock.Whereas Gulia and Wiemer (2019) could discriminate between foreshocks and aftershocks for some sequences characterized by the occurrence of a large foreshock.Spatial variations of the b values are generally investigated by mapping their value on a spatial grid.This allows us to determine the local fluctuation of the stress state.Indeed, this approach has also been used to characterize the stress state on earthquake faults (see among the others Tormann et al. (2014) and references therein).Some authors weight each earthquake based on the distance from the node.This method, first introduced by Tormann et al. (2014), has been modified and applied to several regions of the world by many authors (see among the others Kamer and Hiemer (2015), Taroni et al. (2021), García-Hernández et al. (2021), and Pino et al. (2022)).In all these cases, the most widely used approach is to grid the space into equal-sized cells and select the earthquakes according to certain rules (minimum number of events, maximum distance from the center of the cell, etc.).This approach will cause, in some cases, the overlapping of the cells or, in other cases, the omission of some earthquakes in the node (Godano et al., 2022).This may prevent a formally correct statistical comparison between different cells of the grid.
Due to its inverse relationship with the stress state, the investigation of b for aftershock sequences assumes a critical role in seismic hazard assessment.b value time variations in aftershock sequences have been accurately investigated by Gulia et al. (2018) and Gulia and Wiemer (2019).They carefully consider the Short-Term Aftershock Incompleteness (STAI) (Kagan, 2004).Here we focus on spatial b value variations not considering the effect of STAI, simply because the longer time scale makes STAI effects negligible (see data and methods and appendix for a more detailed discussion).
More recently Godano et al. (2022) introduced a parameter-free method for the gridding of space and b evaluation.This method produces a fully independent mapping of the b value and reduces the number of earthquakes that are omitted.Here, we adopt this method to build b maps on the faults of the major (M > 7) South California earthquakes.Moreover, we evaluate the time interval Δt during which each cell of the grid remains active.More precisely Δt represents the time elapsed from the occurrence of the mainshock and the last event that occurred in the cell.The two parameters allow us to follow the dynamics of the aftershock sequence on the fault.

Data and Method
In our study, we use the relocated South California catalog (Hauksson et al., 2012) which can be downloaded at the website scedc.caltech.edu/research-tools/altcatalogs.html(SCEDC, 2013) focusing our attention on the earthquakes with magnitudes larger than 7, namely Landers 1992, June 28, M W 7; Hector Mine 1999, October 16, M W 7.1; Baja California 2010, April 4, M W 7.2 and Searles Vallet 2019, July 6, M W 7.1.For each earthquake, we chose a fault model, then selected all the on-fault earthquakes from the catalog, and finally apply the Godano et al. (2022) method for evaluating the b maps on the fault using 500 events per cell.In the following, we neglect the occurrence of background activity on the faults.However, this does not imply a loss of generality in our results.Indeed Bottiglieri et al. (2009) showed that it is negligible during aftershock sequences.

Selecting On-Fault Seismicity
We selected the fault geometry using the models provided by the Earthquake Source Model Database at the website http://equake-rc.infoo/srcmodo/(Mai & Thingbaijam, 2014).Specifically, for Landers, we used the fault geometry model proposed by Wald and Heaton (1994), for Hectormine, the fault geometry proposed by Ji et al. (2002), for Baja California, the fault geometry proposed by Mendoza and Hartzell (2013), and for Searles Valley, the fault geometry proposed by Ross et al. (2019).These models appear to be representative of the selected fault geometries, which are very similar for all models on the website.Conversely, differences characterize the slip maps, but they do not influence the method that has been adopted.Finally, we selected all earthquakes occurring within a distance of 3 km from the fault plane as being on-fault seismicity.This value represents the minimum distance that allows the selection of several events sufficiently large for our analysis.Our tests confirmed that using smaller values reduced the number of events available for the analysis to an unacceptable level.Conversely, larger values appear to include off-fault seismic activity.The faulting geometry of Landers and Hector Mine is very complex, and the existing models propose three different fault segments for each earthquake.To select the on-fault seismicity, we consider each fault segment as an independent fault with its associated aftershocks.We then compile all chosen events into a comprehensive catalog for the entire fault, eliminating potential redundancies resulting from intersections between fault segments.Even for Searless Valley, a complex fault geometry has been suggested (Ross et al., 2019), however the complexity is limited to the shallower part of the fault.As a consequence, we decided to adopt a single fault plane of a geometrically complex fault with multiple segments, as suggested by the authors.Finally, the adopted model for Baja California is a single fault segment.We selected the fault geometry using the models provided by the Earthquake Source Model Database at the web-site http://equake-rc.infoo/srcmodo/(Mai & Thingbaijam, 2014).Specifically, for Landers, we used the fault geometry model proposed by Wald and Heaton (1994), for Hectormine, the fault geometry proposed by Ji et al. (2002), for Baja California, the fault geometry proposed by Mendoza and Hartzell (2013), and for Searles Valley, the fault geometry proposed by Ross et al. (2019).These models appear to be representative of the selected fault geometries, which are very similar for all models on the website.Conversely, differences characterize the slip maps, but they do not influence the method that has been adopted.Finally, we selected all earthquakes occurring within a distance of 3 km from the fault plane as being on-fault seismicity.This Earth and Space Science 10.1029/2023EA002933 value represents the minimum distance that allows the selection of several events sufficiently large for our analysis.Our tests confirmed that using smaller values reduced the number of events available for this analysis to an unacceptable level.Conversely, larger values appear to include off-fault seismic activity.The faulting geometry of Landers and Hector Mine is very complex, and the existing models propose three different fault segments for each earthquake.To select the on-fault seismicity, we consider each fault segment as an independent fault and then rejoin the selected aftershocks in a unique fault, avoiding superpositions.Even for Searles Valley, a complex fault geometry has been suggested (Ross et al., 2019), however the complexity is limited to the shallower part of the fault.Consequently, we decided to adopt a single fault plane of a geometrically complex fault with multiple segments, as suggested by the authors.Finally, the adopted model for Baja California is a single fault segment.

Building Independent Cells on Faults
We divided the on-fault seismicity into independent cells by adopting the method proposed by Godano et al. (2022).The method individuates the largest event in the catalog not yet assigned to a cell and builds a cell around the chosen earthquake containing n ± n tol events.The method presents some advantages: (a) no events in the catalog are left unused (b) there are no cells overlapping, (c) the cells are totally independent (allowing statistical comparison of the b values obtained), (d) the method does not require any tuning of the parameters, being the cell size variable and automatically selected.The last of these advantages makes the method very suitable for surveillance systems.Here n = 500 and n tol = 50.When M max M min < 1.5, the cell was discarded from the analysis.Different values of n tol were tested in Godano et al. (2022).The two parameters n and M max M min depend on each other.Indeed, a very large M max M min reduces the number of cells too much and does not allow any interpretation of the results whereas a too small value results in a biased estimation of b.At the same time, a smaller value of n reduces the number of cells respecting the constraint M max M min < 1.5, whereas a very large value significantly reduces the number of cells.
As already showed by Godano et al. (2022) the choice of n = 500 appears to be optimal; however, we have checked that by choosing n = 600, 700 and 800 we do not observe significant differences in the b value maps (see Supporting Information S1).The only difference is represented by the number of cells N f used to evaluate the b value maps (N f vs. n is shown in the Supporting Information S1): for Hector Mine and Searles Valley, the difference is limited to some units; for Baja California N f assumes the same value for each n; and for Lander N f differences reach a maximum value 10 (see Supporting Information S1).

Evaluating the Completeness Magnitude and the b Value
The completeness magnitude M c was evaluated in each cell using the Godano and Petrillo (2022) method.This is based on the observation that the average magnitude 〈M〉 should increase with a magnitude threshold M th .Consequently, the quantity 〈M〉 M th should assume a constant value k for each M th .Of course, this is not true if M th < M c .Godano and Petrillo (2022) evaluated M c as the first M th for which 〈M〉 M th k crosses the zero lines.They estimated the k value as the average 〈M〉 M th in the range M th ∈ [2.5, 4].In this study, we improved their method by adopting a more accurate estimate of k.Here k is evaluated by fitting 〈M〉 M th (M th ) with the function y = e αx + k.An example of such a fit is presented in the Supporting Information S1.For all the earthquakes analyzed, the maps in the SI show a great variability of M c , with the largest range being [1.0,3.6] for Searles Valley, whereas the smallest range [1.4,3.0] is observed in the case of Hector Mine.This reveals that M c does not depend only on the seismic station distribution.Indeed, M c values are significantly different even in very close cells.Fluctuations in a number of factors could influence the M c values: seismic noise, site effects, rock elastic properties and the STAI (Kagan, 2004).The impact of STAI requires a further short discussion here.The decrease in the M c value with the time elapsed from the occurrence of the mainshock has been interpreted as an artifact from the seismic catalog (Kagan, 2004).Indeed, the capability to detect small earthquakes decreases significantly immediately after the occurrence of the mainshock (see Kagan (2004) for a detailed discussion) and is restored after a short time interval depending on the mainshock magnitude.Here, we decided to consider STAI following the approach of Petrillo and Lippiello (2023).Namely, for each earthquake, we evaluate the incompleteness magnitude M c = M main 0.8 ln(t i t main ) 1 (1) Earth and Space Science

10.1029/2023EA002933
where M main is the magnitude of the mainshock, t main is the occurrence time of the mainshock, and t i is the occurrence time of the earthquake under consideration.If the magnitude of this event M i is smaller than M c , the event is discarded from the analysis.The number of discarded events N S is reported in Table 1 for each mainshock analyzed here.As can be seen N S is negligible for almost all mainshocks.The only exception is Searles Valley, where the N S is 22% of the events used for the b value maps.The Δt and b maps for Searles Valley after removing the 71 events in the STAI period are shown in the Supporting Information S1.
The reasons for this observation are two: (a) the procedure of building the cells leaves out many events from the analysis and (b) the STAI is negligible at the time scale considered here.A rough estimate of the omitted events in a cell with Δt = 8 years (see results) is about 6% (see appendix) assuming that: (a) the STAI time interval is approximately 2 days and (b) M c is overestimated to be 1.This value cannot significantly influence our results.Moreover, an overestimation of M c cannot affect the estimation of b, but only its standard deviation, due to the exponential nature of the Gutenberg-Richter law.
After the estimation of M c , we compute the b values only for those cells where condition M max Mc ≥ 1.5 holds true, using the maximum likelihood method (Aki, 1965); whereas the estimation of the b standard deviations σ b uses the method of Shi and Bolt (1982).The choice of M max Mc ≥ 1.5 is derived from the fact that larger values imply a smaller number of cells, whereas smaller values significantly increase the b standard deviation.
The number of events n t selected for each earthquake and the number of events n used for the maps are reported in Table 1.The quantity n t n represents the sum of two numbers: the number of events with magnitude smaller than M c and the number of events discarded because they occur in cells with M max M min < 1.5 or M max M c < 1.5.In the same Table, we report the total number of cells built using the Godano et al. (2022) method N t , the final number of cells used for the evaluation of the b maps N f , the number of cells discarded because M max M min < 1.5 N d1 and the number of cells discarded because M max M c < 1.5 N d2 .Of course, when M max M min < 1.5 we discard 500 ± 50 events for cell.Conversely, when M max M c < 1.5, the number of discarded events due to the application of the criterion is smaller.Let us call the range of this number for cells N r .

Δt Evaluation
The quantity Δt is the time interval between the occurrence of the mainshock and the last event belonging to the cell and represents the duration of the aftershock sequence in the cell.Of course, the maximum value that Δt can assume is the duration of the catalog minus the time of the mainshock occurrence.However, such a value is never taken by Δt for the events investigated here (see results).Δt depends strongly on the fault complexity.Consequently, it represents an indication of how close the cell is to some heterogeneity of the fault.

Results and Discussion
For each earthquake analyzed here, we present the maps of b and Δt in the main text, whereas the M c and σ b maps are reported in the SI.In each figure, we show the fault plane as dashed gray lines.The green star represents the mainshock, whereas the black stars are the on-fault aftershocks with magnitudes greater than 4.5.
The resulting maps for each earthquake are discussed separately below.

Landers
The faulting process of this earthquake is very complex and involves three different fault segments, as shown in Figure 1.The Δt distribution on the fault reveals that almost all the cells remain active (in the sense that aftershocks occur therein) for less than 8 years and only four cells are active for Δt ≥ 18 years.They are located just at the edges of the fault system, where two fault segments cross each other.The smallest b value (0.7) is observed at the southern margin of the fault, which represents a location of high stress concentration.This is compatible with the presence of a barrier (namely a hard fault patch) blocking the faulting process at the southern margin of the Note.The total number of cells built using the Godano et al. ( 2022) method N t , the final number of cells used for the evaluation of the b maps N f , the number of cells discarded because M max M min < 1.5 N d1 and the number of cells discarded because M max M c < 1.5 N d2 .N r represents the range of the discarded number of events in cells with M max M c < 1.5 after the evaluation of M c .

Earth and Space Science
10.1029/2023EA002933 CONVERTITO ET AL.
fault, as already observed by Wiemer and Wyss (1997).Conversely, there are only eight cells with b > 1.3 (b max = 1.5)where the stress is lower.

Hector Mine
Even if the faulting process for this earthquake appears to be very complex, being composed of three different segments (Figure 2), the distribution of the b values in the cells is smoother when compared to the Landers earthquake.Indeed, we find b values in the range [0.8,1.2].However, the smallest values are at the fault ridges and at the intersection of two fault segments.Consequently, it could be possible to interpret these values as due to the presence of barriers or kinks in the faults.In general, the Δt assumes values smaller than 8 years, and only for two cells do we observe Δt ≈ 20 years at the northern edge of the fault.

Baja California
In this case, the faulting process is very simple and occurs on a planar fault (Figure 3).The small number of onfault aftershocks and, consequently, the small number of statistically significant cells as well as the small values of Δt reveals that the on-fault aftershock activity is largely controlled by the complexity of the faulting process.For this earthquake, we were not able to associate the presence of barriers with the b values because of the very small number of cells.

Searless Valley
The fault process for this earthquake also appears to be very simple, and we obtain a homogeneous distribution of both the b values, in the range [0.5,1.1], and the Δt whose maximum value is 0.5 years (Figure 4).
In all cases, the absence of a correlation between Δt and b reveal that the duration of the aftershock activity does not depend on the stress level.The same observation can be made for the occurrence of the largest aftershocks whose location does not correspond to cells with small b values.

Comparison With Previous Results
To the best of our knowledge, the on-fault b values previously evaluated concern Landers and Hector Mine earthquakes.The first one focused the researchers' attention because it changed the seismicity rate in a large part of California (Wiemer & Wyss, 2000).The temporal and spatial variations of the b value on the Landers fault have been estimated using different approaches (Tormann et al., 2014;Wiemer & Wyss, 2002).They found the highest Their results are compatible with ours (Figure 1), with the exception that we did not observe significant aftershocks in the nucleation zone.Such a result can be easily explained by observing that a large part of the stress released during the mainshock occurs in the nucleation zone.Consequently, the probability of observing aftershocks in this zone is negligible.Tormann et al. (2014) observed that the b value in the area of the Hector Mine earthquake remained almost constant and close to 1, with a tiny decrease in the very shallow values and an increase below, immediately before the event.After the Hector Mine earthquake, the b value increases on the fault concerned, and an area of small b value is evident in the shallow eastern sector.These observations are compatible with our results (Figure 2).In particular, the small b value highlights the presence of barriers to propagation at the edges of the fault.
The differences between the Tormann et al. ( 2014) observations and ours can be attributed to the different methods.Indeed, many methods exist for the estimation of the b value variations on a fault: the fixed radius method, the N-nearest earthquakes method, and the Distance Exponential Weighted (DEW) method.The first two are implemented in Z-map (Wiemer, 2001), the most used and tested program for b value estimation, and the third is well described in Tormann et al. (2014).The third method takes advantage of the independence from the chosen distance from the fault for earthquakes to be included in the b value estimation, which is instead an input parameter for the other two methods.In addition, with the DEW method, a parameter must be set that is the gradient of the distance weight decay.
Our method sits in the middle between these approaches as it needs the definition of the maximum distance of the earthquakes from the fault, but it has the advantage of giving a statistically independent value of b (Godano et al., 2022) and shows unsmoothed b value maps.

Discussion and Conclusions
The duration of the aftershock sequence, the b distribution, and the number of on-fault aftershocks appear to strongly depend on the geometrical complexity of the fault.We observe that Lander and Hector Mine present a more complex fault composed of three segments (Figures 1 and 2), and in both cases the b distribution is more heterogeneous when compared to Baja California and Searles Valley (Figures 3 and 4).
Interestingly, for Landers, Hector Mine, and Searles Valley, we can perform a correlation between the slip on the fault and the cell distribution.Unfortunately, such analysis cannot be performed for Baja California because of the small number of cells selected by the method used here.Landers exhibits an absence of cells in the fault zone of maximum slip (Wald & Heaton, 1994).Note that the absence of cells can be caused by a true absence of earthquakes or by a small magnitude seismicity due to M max M min < 1.5 or to a small magnitude range M max M c < 1.5.In all cases, the maximum slip zone is characterized by the absence of significant seismicity.This is a strong indication that the stress was almost completely released during the mainshock faulting process.Conversely, for the Hector Mine significant seismicity (characterized by small b value and high stress but with a small Δt) occurs at the left of the sinistral fault whereas no earthquakes occur at the right of the where we observe the maximum slip in both areas during the mainshock faulting (Ji et al., 2002).A possible scenario is that the faulting nucleates at the left of the kink and then proceeds beyond.As a result, all the stress has been released but the Coulomb stress of this area reactivates the part of the fault to the left of the kink for a short time interval.For Searles Valley earthquakes surrounding the zone of maximum slip (Ross et al., 2019) have small b and Δt values, suggesting that the stress has been released during the faulting process and increased in the part of the fault not slipped during the mainshock faulting.
Our results confirm that the b parameter is a good stress meter and can be used to characterize the faulting process.As a concluding remark, we would like to observe that the use of the Godano et al. (2022) method allows a better resolution of the b values on the fault due to their independence and the small number of earthquakes lost in the griding process.

Figure 1 .
Figure 1.The map of the Δt and b values on the Landers earthquake fault.The fault plane is shown by dashed gray lines.The green star represents the mainshock, whereas the black stars are the on-fault aftershocks with magnitudes greater than 4.5.

Figure 2 .
Figure 2. The map of the Δt and b values on the Hector Mine earthquake fault.The fault plane is shown by dashed gray lines.The green star represents the mainshock, whereas the black stars are the on-fault aftershocks with magnitudes greater than 4.5.

Figure 3 .
Figure 3.The map of the Δt and b values on the Baja California earthquake fault.The fault plane is shown by dashed gray lines.The green star represents the mainshock, whereas the black stars are the on-fault aftershocks with magnitudes greater than 4.5.

Figure 4 .
Figure 4.The map of the Δt and b values on the Searles Valley earthquake fault.The fault plane is shown by dashed gray lines.The green star represents the mainshock, whereas the black stars are the on-fault aftershocks with magnitudes greater than 4.5.

Table 1
The Number of Events Selected for Each Earthquake Here Analyzed and the Number of Events Used for the b Map Estimation