Quantifying the probability and uncertainty of multiplestructure rupture and recurrence intervals in Taiwan
 ^{1}Department of Earth Sciences, National Central University, Taoyuan, 32001, Taiwan
 ^{2}EarthquakeDisaster & Risk Evaluation and Management (EDREaM) Center, National Central University, Taoyuan, 32001, Taiwan
 ^{1}Department of Earth Sciences, National Central University, Taoyuan, 32001, Taiwan
 ^{2}EarthquakeDisaster & Risk Evaluation and Management (EDREaM) Center, National Central University, Taoyuan, 32001, Taiwan
Abstract. This study identifies fault pairs with potential for simultaneous rupture in a coseismic period based on a physicsbased model, and proposes a set of formulas to evaluate their recurrence intervals and uncertainties. To assess the potential for a multiplefault rupture, we calculated the probability of stress triggering between active faults. We assumed that a multiplefault rupture would occur if two faults could trigger each other by enhancing the plane with thresholds of a stress increase and the distance between the faults. To estimate the recurrence intervals for multiplefault ruptures, we implemented a statisticsbased model in which the slip rate could be partitioned based on the earthquake magnitudes of the individual fault and multiplefault ruptures. Due to a larger characteristic magnitude and a larger displacement of the multiplefault rupture, its recurrence interval could be longer. Therefore, application of the multiplefault rupture could lead to an increase in seismic hazard in a long return period, which would be crucial for the safety evaluation of infrastructures, such as nuclear power plants and dams.
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ChiehChen Chang et al.
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RC1: 'Comment on nhess202246', Jack Williams, 27 Feb 2022

AC1: 'Response to Reviewer #1 Jack Williams', ChungHan Chan, 14 May 2022
We greatly appreciate the reviewer’s insightful comments and have revised our manuscript, nhess202246, entitled, “Quantifying the probability and uncertainty of multiplestructure rupture and recurrence intervals in Taiwan,” accordingly. Below, we have quoted the comments in italics and provided our detailed responses. All the changes are underlined in the revised manuscript.
1.) Description of model
The key innovation of this study is described in Section 3.1 where it is outlined how the area and slip rate (i.e., a moment rate) of two different seismogenic structures can be combined with a GR relationship to determine the recurrence interval of an earthquake that ruptures both structures. As far as I can tell, there is nothing inherently wrong with the approach itself, however, I have several recommendations for how the presentation of this model could be improved.
We appreciate the reviewer’s very helpful recommendations. We considered the reviewer’s comments and responded in the following.
Immediately after equations 8 and 9 (Line 129), the meaning 𝐷̇𝐿1 (original L1 slip rate measurement) is given, but it is 𝐷̇𝐿1′ (slip rate for L1 single structure events) that is used in these equations. I would also present the equations for C1 (partitioning coefficient between 𝐷̇𝐿1+𝐿2𝐿1 and 𝐷̇𝐿1′, currently eqs. 10 and 11) and 𝐷̇𝐿1′ (currently eqs. 12 and 13) before the equation for 𝐷̇𝐿1+𝐿2𝐿1 (slip rate of L1 in L1+L2 events) given that you need these parameters to calculate 𝐷̇𝐿1+𝐿2𝐿1.
To clearly describe our algorithm for evaluating recurrence interval of multiplestructure ruptures, we first introduced the slip rate partitioned to individual structure ruptures (equations 8 and 9), followed by the obtained partitioned rates (equations 10 and 11). By combining them, the slip rate partitioned to the multiplestructure rupture from the original structures can be obtained (described in lines 123140).
I appreciate that the authors use the Hsinhua and Houchiali faults to provide an example of how their workflow is applied. However, showing the application of each equation to each structure in the text can get repetitive. I would suggest using a table to illustrate these equations, with a column for each structure. Another table could also be used for description of the model where >2 structures are considered (i.e., the example of the Chiayi, Meishan, and Tainan structures in Section 3.2).
In the previous manuscript, this example is provided to demonstrate the procedure of the workflow. To simplify the description of the calculation, this example has been removed.
When using the examples from the TEM, the values are provided to a high, and probably unjustified level of specificity (e.g. slip rates to 0.01 mm/yr, source areas to 0.01 km2, recurrence intervals to 1 year). I suggest rounding these values to a level appropriate with the uncertainty of this analysis.
We followed the reviewer’s comment and revised Table 1 accordingly. Now the slip rate and slip area are rounded to one decimal place and the nearest whole number, respectively. Note that we keep recurrence intervals to 1 year, since some structures (e.g., the Milun fault) obtain short recurrence intervals (<100 years).
Finally, the Wells and Coppersmith (1994) scaling relationships are increasingly out of date given that we now have nearly 30 more years’ worth of observed earthquakes to refine these relationships. I would recommend that either a more up to date set of scaling relationships is used (e.g., Leonard 2010, Thingbaijam et al 2017), or a sensitivity analysis is made to see if using the updated scaling relationships changes the model outcomes.
To validate the sensitivity of our procedure to scaling, we also implemented alternative relationships proposed by Yen and Ma (2011), who investigated the rupture parameters of the earthquakes mainly from the Taiwan orogenic belt.
Based on this relation, recurrence intervals for each multiplestructure rupture pairs were evaluated (Table 5). Comparing these with those obtained by Wells and Coppersmith’s relations, shorter recurrence intervals were obtained, especially for those with larger magnitude. These results can be attributed to a smaller average displacement obtained for a large event that led to a shorter recurrence interval for the multiplestructure rupture (based on equation 17). Note that although the scaling relations proposed by Wells and Coppersmith (1994) have been questioned by many modern models, especially for large megathrusts, Wang et al. (2016^{b}) concluded similar maximal magnitude of each seismogenic structure estimated from the relations of Wells and Coppersmith (1994) and Yen and Ma (2011). We provided more detailed descriptions in lines 214223, 292298.
Multistructure earthquakes are considered here only in terms of static Coulomb stress triggering between neighbouring faults. However, it is worth acknowledging in Section 4.2 that multistructure earthquakes may also be generated by dynamic stress triggering from seismic waves (e.g., Brodsky and van der Elst 2014, Ulrich et al 2018). I think this may what is being discussed at Line 280 (?), though note the reference is to a manuscript (Jiao et al 2020) that was not accepted for publication.
We followed the reviewer’s comment and indicated dynamic models could also constrain the behaviors of multiplestructure ruptures (lines 242245). Note that the paper by Jiao et al. has been published in 2022.
A key assumption in this study is that the magnitudefrequency distribution (MFD) of events along a single multistructure systems follow a GR scaling. Although that is certainly possible, one could also argue that at the scale of a single multistructure system, the MFD follows a characteristic shape (Youngs and Coppersmith 1984; Hecker et al 2013; Stirling and Zungia 2017), or that the MFD is neither characteristic nor GR (Geist and Parsons 2019; Page et al 2021). In either case, a deviation from a GR scaling will affect the recurrence intervals calculated through this model.
We are aware of the importance of the magnitudefrequency distribution (MFD) on a singlestructure rupture, and the MFD could be in various forms, including the GutenbergRichter law and the characteristic earthquake model. In this study, we evaluated the rupture recurrence interval as the ratio of slip of a characteristic earthquake (with maximum magnitude of the structure) and slip rate based on the assumption proposed by the TEM seismogenic structure database and the TEM PSHA2020. Note that this factor could be replaced by other magnitudefrequency distributions since the recurrence interval of the multiplestructure rupture in our procedure is based on slip rate partitioned from individual structure ruptures (shown as equations 89, 14, 18, and 20). We provided more detailed descriptions in lines 101104, 299307.
This model should also be discussed in the context of other studies that have attempted to incorporate multistructure ruptures in PSHA. For example, there are many studies that divide mapped multistructure systems into smaller subfault scale segments, and then essentially allow ruptures to ‘float’ across theses smaller segments in such a way that they fit a regional MFD target (Field et al 2014; 2021; Chartier et al 2019; Geist and Parsons 2019). These studies are therefore distinct from the model described here, which is quite prescriptive about the number of configurations that structures in the TEM can rupture in (i.e., as single or multistructure events only, and no events may be smaller than a single structure). It would benefit this study if the pros and cons of these different techniques could be discussed in Section 4.3.
Based on the assumption of the TEM PSHA2020, every rupture on a seismogenic structure results in a characteristic earthquake (with maximum magnitude of the structure), that is, small earthquakes (with magnitude smaller than the maximum magnitude of the structure) are attributed to shallow background sources. Following this assumption, we did not consider ruptures on small segments of a structure.
Minor Comments
Lines 721: The abstract does not mention that this study is using faults incorporated into the Taiwan Earthquake Model to perform this analysis. Suggest revise, Line 11 could be revised to:
‘……the probability of Coulomb stress triggering between seismogenic structures included in the Taiwan Earthquake Model.’
We followed the reviewer’s comment and revised the text accordingly.
Lines 6468: What value is used for the effective coefficient of friction (μ‘) in the Coulomb stress modelling?
We first assumed a fixed μ‘ of 0.4. To quantify deviation on determining multiplerupture pairs, we further considered µ’=0.2 and 0.5, the boundaries of its reasonable range determined from focal mechanisms in Taiwan. Considering the stress threshold of ∆CFS≥0.1 bar and a distance threshold of 5 km, the potential paired structures were identified (Table 6). The results suggest slight differences within the reasonable effective friction coefficient (lines 5456, 259267).
Line 115: These scaling relationships between magnitude and rupture area are presumably from Wells and Coppersmith (1994)? If so, they should be cited as such (though also see major comment #1)
We followed the reviewer’s comment and cited the reference accordingly.
Line 144: Replace ‘integrating’ with ‘combining,’ to avoid any connotations that you are actually performing an integration in these equations.
We followed the reviewer’s comment and revised the text accordingly.
Lines 220221 (and 335): When referring to the Kaikōura earthquake, reference should be made to Hamling et al (2017). This is the original reference to this event and written by authors who made the primary observations of this multifault earthquake.
We followed the reviewer’s comment and cited the reference accordingly.
Line 258: I think there is a typo here for describing the numeric value if the Hukou and Hsinchu fault recurrence intervals as ‘4.4 and 5.3’?
We have revised the text as “their recurrence intervals become 4.4 and 5.3 times, respectively, longer than the cases without considering multiplestructure ruptures” (lines 212213).
Figures: Figure 1 presents only a generic case of Coulomb stress changes around a fault. I would recommend also including a figure to show an example of this stress modelling from faults in the TEM. Maybe using the example of faults that are described further in Section 3.1?
This figure has been removed.
 AC6: 'The annotated manuscript is available', ChungHan Chan, 14 May 2022

AC11: 'The revised manuscript with annotation is available online', ChungHan Chan, 14 May 2022
The revised manuscript with annotation is available online in the following link:
https://editor.copernicus.org/index.php?_mdl=msover_md&_jrl=7&_lcm=oc108lcm109w&_acm=get_comm_sup_file&_ms=101260&c=224615&salt=4901848291507617197
or at AC5: 'Annotated manuscript'

AC1: 'Response to Reviewer #1 Jack Williams', ChungHan Chan, 14 May 2022

RC2: 'Comment on nhess202246', João Fonseca, 03 Mar 2022
Chang et al.´s manuscript address an interesting and relevant problem: three decades after the observation that two separate faults can interact in such ways that the rupture of one may trigger the rupture of the other (Stein et al., 1992), how can we incorporate this phenomenon in hazard assessment? The authors use a database of seismogenic faults in Taiwan to investigate the likelihood that such stress triggering of earthquakes could take place over a given period of time, and to quantify its impact on seismic hazard.
The authors apply an analytical procedure inspired in Chan et al. (2020), which stems from the notion that when two faults interact each one gives part of its slip rate to the multiplerupture process, while retaining the remaining slip rate to its individualrupture process. Starting from this basic notion – the “partitioned slip rates” of Chan et al. (2020)  the authors use Kanamori’s (1977) definition of moment magnitude, the definition of seismic moment, Wells and Coppersmith’s (1994) scaling relations of moment magnitude with rupture area and the GutenbergRichter relation to obtain return periods for multiple ruptures and for individual ruptures on each fault.
Key to the authors' reasoning is their assumption that the slip rate of each fault is partitioned between individual and multiple ruptures according to a “partitioned rate” given by their equations 10 and 11, which include "the magnitude of the multiplestructure rupture” (line 135) and “the displacement of the multiplerupture structure” (line 136). This terminology reflects the fact (not explained in the manuscript) that the catalog used in the study considers characteristic ruptures only. The expression for C ifeatures the bvalue of the GutenbergRichter relation. The authors present the expression for C as a logical conclusion of the GutenbergRichter relation, although I was not able to follow that logic. I was able to trace the definition of the partitioned rate C to Chan et al. (2020), but there too it was introduced without an explanation.
The authors estimate a return period for a multiple rupture in a pair of faults as R_{12}=D_{12}/D_{12}', where D_{12} is a displacement associated with the joint rupture of the two faults and D_{12}' is a slip rate associated with the joint rupture of the two faults. To obtain D_{12} the authors use the definition of seismic moment inserting for A the sum of the two fault areas and estimating the seismic moment from a value of magnitude inferred from A using the Wells and Coppersmith (1994) scaling relations. To obtain the multiplerupture sliprate the authors assume that D_{12}'=D^{12}_{1}'+D^{12}_{2}', where D^{12}_{k}' is the part of the slip rate of faullt k that takes place through joint rupture (estimated with the partitioned rates discussed above). It is important to inquire into the physical meaning of these quantities. The estimate of the multiplerupture displacement is formally correct, albeit highly convoluted. But the estimate of the multiplerupture slip rate through a sum defies logic, in my view (why sum slip rates interesting separate faults?) Also, as pointed out above, each parcel relies on a coefficient that was not sufficiently explained.
In section 3.2 the authors enlarge their approach to include more than two faults in interaction, increasing the complexity while inheriting the obscurity from the previous section.
In sections 3.3 and 4, the authors discuss some implications of their analysis for seismic hazard. Around line 245, the authors conclude that the possibility of multiplerupture earthquakes reduces the hazard at the shorter return periods while increasing it at longer return periods. In line 255, the authors observe that “structures that pair with several cases of multiplestructure ruptures might be difficult to rupture solely”. These observations are so clearly at odds with empirical evidence – which points to singlefault rupture as the dominant contributor to hazard – that they should be regarded as indicating flaws of the approach.
The authors base their approach on a simplified view of stress transfer between faults: they ignore dynamic effects, porefluid effects and – surprisingly in view of published evidence – restrict the range of stress transfer to 5km. Although the title promised a quantification of the uncertainties, very little is done to quantify the errors that derive from such simplifications. In line 263 the authors state that their approach is a physicsbased one. Unfortunatelly, it seems to have strayed strongly from the geological reality of earthquake generation. The authors recognize, to their credit, that the “analysis could be further improved through better understanding seismogenic structures” (line 278). I would take this conclusion even further and say that the analysis needs to be reformulated starting with a better understanding of seismogenic processes. For example, exploring empirical evidence of the occurrence and characteristics of multiplerupture earthquakes in the available databases, in order to be able to subject their model to a reality check.
In the present stage of development, I regret to conclude that I don’t consider this research ready for publication.
Lisbon, March 3, 2022
Joao Fonseca

AC2: 'Response to Reviewer #2 João Fonseca', ChungHan Chan, 14 May 2022
We greatly appreciate the reviewer’s insightful comments and have revised our manuscript, nhess202246, entitled, “Quantifying the probability and uncertainty of multiplestructure rupture and recurrence intervals in Taiwan,” accordingly. Below, we have quoted the comments in italics and provided our detailed responses. All the changes are underlined in the revised manuscript.
The authors apply an analytical procedure inspired in Chan et al. (2020), which stems from the notion that when two faults interact each one gives part of its slip rate to the multiplerupture process, while retaining the remaining slip rate to its individualrupture process. Starting from this basic notion – the “partitioned slip rates” of Chan et al. (2020)  the authors use Kanamori’s (1977) definition of moment magnitude, the definition of seismic moment, Wells and Coppersmith’s (1994) scaling relations of moment magnitude with rupture area and the GutenbergRichter relation to obtain return periods for multiple ruptures and for individual ruptures on each fault. Key to the authors' reasoning is their assumption that the slip rate of each fault is partitioned between individual and multiple ruptures according to a “partitioned rate” given by their equations 10 and 11, which include "the magnitude of the multiplestructure rupture” (line 135) and “the displacement of the multiplerupture structure” (line 136). This terminology reflects the fact (not explained in the manuscript) that the catalog used in the study considers characteristic ruptures only. The expression for C ifeatures the bvalue of the GutenbergRichter relation. The authors present the expression for C as a logical conclusion of the GutenbergRichter relation, although I was not able to follow that logic. I was able to trace the definition of the partitioned rate C to Chan et al. (2020), but there too it was introduced without an explanation.
I replied this comment through three aspects, algorithm description, scaling relation, and assumption of characteristic ruptures, detailed below.
To clearly describe our algorithm for evaluating the recurrence interval of multiplestructure ruptures, we first introduced the slip rate partitioned to individual structure ruptures (equations 8 and 9), followed by the obtained partitioned rates (equations 10 and 11). By combining them, the slip rate partitioned to the multiplestructure rupture from the original structures could be obtained (described in lines 122140).
Although the scaling relations proposed by Wells and Coppersmith (1994) have been questioned by many modern models, especially for large megathrusts, Wang et al. (2016^{b}) concluded a similar maximal magnitude of each seismogenic structure estimated from the relations of Wells and Coppersmith (1994) and Yen and Ma (2011), obtained from regressions of the rupture parameters of the earthquakes mainly from the Taiwan orogenic belt. Besides, to validate the sensitivity of our procedure to scaling, we implemented alternative relationships proposed by Yen and Ma (2011). Based on this relation, recurrence intervals for each multiplestructure rupture pairs were evaluated (Table 5). Comparing these with those obtained by Wells and Coppersmith’s relations, shorter recurrence intervals were obtained, especially for those with larger magnitude. These results can be attributed to a smaller average displacement obtained for a large event that led to a shorter recurrence interval for the multiplestructure rupture (based on equation 17). We provided more detailed descriptions in lines 214223, 292298.
The estimate of the multiplerupture displacement is formally correct, albeit highly convoluted. But the estimate of the multiplerupture slip rate through a sum defies logic, in my view (why sum slip rates interesting separate faults?) Also, as pointed out above, each parcel relies on a coefficient that was not sufficiently explained.
To clearly describe our algorithm for evaluating the recurrence interval of multiplestructure ruptures, we have modified the manuscript to first introduce the slip rate partitioned to individual structure ruptures (equations 8 and 9), followed by the obtained partitioned rates (equations 10 and 11). By combining them, the slip rate partitioned to the multiplestructure rupture from the original structures can be obtained (described in lines 122140).
The estimate of the multiplerupture slip rate through a sum is based on the assumption that the slip of an earthquake is equal to the cumulative slip during an interseismic period. Since the slip of a multiplestructure rupture is the result of contributions from different structures, we sum the slip rates contributed from the individual structures.
In section 3.2 the authors enlarge their approach to include more than two faults in interaction, increasing the complexity while inheriting the obscurity from the previous section.
Earthquakes could be attributed to multiple (more than three) structures, for example, the 2010 El MayorCucapah, US, earthquake; the 2016 Mw7.8 Kaikōura, New Zealand, earthquake. The procedure we proposed in Section 3.2 could quantify the return period of these earthquakes.
In sections 3.3 and 4, the authors discuss some implications of their analysis for seismic hazard. Around line 245, the authors conclude that the possibility of multiplerupture earthquakes reduces the hazard at the shorter return periods while increasing it at longer return periods. In line 255, the authors observe that “structures that pair with several cases of multiplestructure ruptures might be difficult to rupture solely”. These observations are so clearly at odds with empirical evidence – which points to singlefault rupture as the dominant contributor to hazard – that they should be regarded as indicating flaws of the approach.
The description mentioned here is based on the comparison between models with and without multiplestructure ruptures. That is, the return period of a seismogenic structure could be longer if a part of its coupling rate will contribute to the multiplestructure rupture. Note that based on our procedure, a shorter return period is expected for a rupture on one individual structure than for a multiplestructure rupture. For example, we obtained a return period of 6,640 and 11,953 years for the Hsinchu fault and multiplestructure rupture of the Hukou fault and Hsinchu fault, respectively.
The authors base their approach on a simplified view of stress transfer between faults: they ignore dynamic effects, porefluid effects and – surprisingly in view of published evidence – restrict the range of stress transfer to 5km. Although the title promised a quantification of the uncertainties, very little is done to quantify the errors that derive from such simplifications. In line 263 the authors state that their approach is a physicsbased one. Unfortunatelly, it seems to have strayed strongly from the geological reality of earthquake generation. The authors recognize, to their credit, that the “analysis could be further improved through better understanding seismogenic structures” (line 278). I would take this conclusion even further and say that the analysis needs to be reformulated starting with a better understanding of seismogenic processes. For example, exploring empirical evidence of the occurrence and characteristics of multiplerupture earthquakes in the available databases, in order to be able to subject their model to a reality check.
We followed the reviewer’s comment and included some more discussion on various physicsbased components, including effective coefficient of friction (Table 6, lines 259267), rake angle rotation (Table 8, lines 282285), stress threshold of ∆CFS (Table 3, lines 8994, 268272), and distance threshold (Tables 3 and 7, lines 8994, 268272, 273281). Note that we explained our model without implementing a poroelastic assumption, since previous studies (e.g., Chan and Stain, 2009) concluded that the differences in their results were trivial for assuming reasonable values of Skempton’s coefficients (in between 0.5 and 0.9) and dry friction (0.75). Our approach indicated various rupture pairs and quantified uncertainties. These outcomes could be incorporated into a probabilistic seismic hazard assessment through a logic tree.
In the present stage of development, I regret to conclude that I don’t consider this research ready for publication.
We appreciate the reviewer’s very helpful comments. We hope the adjustments we have made accordingly to the manuscript meet the standards of Natural Hazards and Earth System Sciences and have made the manuscript to now ready for publication.
 AC7: 'The annotated manuscript is available', ChungHan Chan, 14 May 2022

AC10: 'The revised manuscript with annotation is available online', ChungHan Chan, 14 May 2022
The revised manuscript with annotation is available online in the following link:
https://editor.copernicus.org/index.php?_mdl=msover_md&_jrl=7&_lcm=oc108lcm109w&_acm=get_comm_sup_file&_ms=101260&c=224615&salt=4901848291507617197
or at AC5: 'Annotated manuscript'

AC2: 'Response to Reviewer #2 João Fonseca', ChungHan Chan, 14 May 2022

RC3: 'Comment on nhess202246', Anonymous Referee #3, 05 Mar 2022
Huang et al., in the manuscript "Quantifying the probability and uncertainty of multiplestructure rupture and recurrence intervals in Taiwan" presents a new approach by integrating the physicsbased model (static Coulomb stress change) and statistic model (GutenbergRichter law) to evaluate the earthquake recurrence time for the possible multiplerupture scenario. According to their assumption, multirupture only occurs if the stress transfer on the nearby fault reaches a certain value, and the slip rate of the multiplerupture structure is the sum of the associated slip rate in related ruptures. Although I acknowledge this topic as a valuable contribution in the field of hazard assessment, however, this current manuscript needs improvements, especially I am still not clear about how the author partitioned the slip rates between different ruptures.
The structure of the description.
I think section 3 is the core of the methodology in this study, as far as I can tell this study use simple equations, but the description makes it extremely difficult to follow.In general, I think the whole section of 3.1 and 3.2 should be reformulate, for example:
Equation (2),D^dot represents the slip rate, dose this slip rate indicates the longterm slip rate obtained from other measurements?Equation (7), the author used the MwMo scaling law by Kanamori (1977), but the equation in the manuscript is from Hanks and Kanamori (1979) with the unit of dynecm.
Equation (8) and (9), there appear two parameters D_L1’ and D_L2’ with no explanations until equation (12) and equation (13).
Equation (10), dose the ML1 indicates the maximum magnitude in L1 ?
D_L1+L2 is the displacement of the multiplestructure rupture, dose this means D_L1+L2 = D_L1 + D_L2? More practical parameter annotation should be carefully addressed.Equation (14), this equation is hard to follow, in Line 146 : the sum of the slip rates for the multiplestructure…. I don’t understand what is the sum of the slip rates for the multiplestructure? and this statement is not correspond to the equation (14).
The discussion:
The author took 1906 Meishan earthquake as an example, they argued that closedby Chiayi frontal structure also ruptured during the coseismic period because liquefaction took place on the west of the Meishan fault, however, I think this statement is littlebit weak because liquefaction could occur when the stress is perturbated through seismic wave propagation from the mainshock. Also, I got confused when reading the line from 286 to 288, dose the author really hints that Meishan earthquake is initiated on the Chiayi frontal structure?For model uncertainty, this sensitivity test is focus only on the rake angles for estimating the Coulomb stress change, I was wondering what if they change the friction coefficient? Friction coefficient also plays an important role on evaluating the stress impart from the mainshock, especially recent studies suggest that friction coefficient is depth dependence (i.e., Carpenter et al., 2012,2015). Besides the Coulomb stress model, GR law also make a strong contribution on this approach, I am wondering if they consider different type of GR law will change the result significantly (for example the truncated model)?
minor comments:
Line 131, show in equation 1 > equation 3
Line 157, what is characteristic earthquake means? rupture or slip or magnitude?
Line 159~ , The author addresses the exact value of each parameter very carefully, but I do think those repetitive equations and number should be removed and only use a simple table to present.
Line 284, missing the ID for Chiayi frontal structure

AC3: 'Response to Reviewer #3 Anonymous Referee', ChungHan Chan, 14 May 2022
We highly appreciate the reviewer’s insightful comments and have revised our manuscript, nhess202246, entitled, “Quantifying the probability and uncertainty of multiplestructure rupture and recurrence intervals in Taiwan,” accordingly. Below, we have quoted the comments in italics and provided our detailed responses. All the changes are underlined in the revised manuscript.
Huang et al., in the manuscript "Quantifying the probability and uncertainty of multiplestructure rupture and recurrence intervals in Taiwan" presents a new approach by integrating the physicsbased model (static Coulomb stress change) and statistic model (GutenbergRichter law) to evaluate the earthquake recurrence time for the possible multiplerupture scenario. According to their assumption, multirupture only occurs if the stress transfer on the nearby fault reaches a certain value, and the slip rate of the multiplerupture structure is the sum of the associated slip rate in related ruptures. Although I acknowledge this topic as a valuable contribution in the field of hazard assessment, however, this current manuscript needs improvements, especially I am still not clear about how the author partitioned the slip rates between different ruptures.
To clearly describe our algorithm for slip rate partitioning, we revised our procedure to first introduce the slip rate partitioned to individual structure ruptures (equations 8 and 9), followed by the obtained partitioned rates (equations 10 and 11). By combining them, the slip rate partitioned to the multiplestructure rupture from the original structures could be obtained (described in Lines 123140).
We hope the present version of the manuscript meets the standards of Natural Hazards and Earth System Sciences and is now ready for publication.
The structure of the description.
I think section 3 is the core of the methodology in this study, as far as I can tell this study use simple equations, but the description makes it extremely difficult to follow. In general, I think the whole section of 3.1 and 3.2 should be reformulate, for example:
Equation (2),D^dot represents the slip rate, dose this slip rate indicates the longterm slip rate obtained from other measurements?
The slip rate (_{L1}, shown in equation 2) is obtained from the TEM seismogenic structure database (Table 1).
To clearly describe our algorithm for the recurrence interval of multiplestructure ruptures, especially for slip rate partitioning, we modified Section 3 and hope the current version achieves the desired clarity (lines 97190).
Equation (7), the author used the MwMo scaling law by Kanamori (1977), but the equation in the manuscript is from Hanks and Kanamori (1979) with the unit of dynecm.
We thank the reviewer highly for having identified this oversight in our paper. We have revised the manuscript accordingly and simplified equation 7.
Equation (8) and (9), there appear two parameters D_L1’ and D_L2’ with no explanations until equation (12) and equation (13).
Equation (10), dose the ML1 indicates the maximum magnitude in L1 ? D_L1+L2 is the displacement of the multiplestructure rupture, dose this means D_L1+L2 = D_L1 + D_L2? More practical parameter annotation should be carefully addressed.
Equation (14), this equation is hard to follow, in Line 146 : the sum of the slip rates for the multiplestructure…. I don’t understand what is the sum of the slip rates for the multiplestructure? and this statement is not correspond to the equation (14).
To clearly describe our algorithm for evaluating the recurrence interval of multiplestructure ruptures, we first introduced the slip rate partitioned to individual structure ruptures (equations 8 and 9), followed by the obtained partitioned rates (equations 10 and 11). By combining them, the slip rate partitioned to the multiplestructure rupture from the original structures could be obtained (described in lines 123140).
The author took 1906 Meishan earthquake as an example, they argued that closedby Chiayi frontal structure also ruptured during the coseismic period because liquefaction took place on the west of the Meishan fault, however, I think this statement is littlebit weak because liquefaction could occur when the stress is perturbated through seismic wave propagation from the mainshock.
To better illustrate the rupture behavior of the Maishan earthquake, we provided evidence such as the larger magnitude than the characteristic magnitude of the Meishan fault, the focal mechanism of oblique thrust faulting being oriented in the northeast–southwest direction, and the large ground shaking with liquefaction that took place to the west. All infer the Chiayi frontal structure might rupture simultaneously.
Also, I got confused when reading the line from 286 to 288, dose the author really hints that Meishan earthquake is initiated on the Chiayi frontal structure?
The description of the simplified Coulomb stress change model has been removed.
For model uncertainty, this sensitivity test is focus only on the rake angles for estimating the Coulomb stress change, I was wondering what if they change the friction coefficient? Friction coefficient also plays an important role on evaluating the stress impart from the mainshock, especially recent studies suggest that friction coefficient is depth dependence (i.e., Carpenter et al., 2012,2015). Besides the Coulomb stress model, GR law also make a strong contribution on this approach, I am wondering if they consider different type of GR law will change the result significantly (for example the truncated model)?
We followed the reviewer’s comment and discussed the impact of the friction coefficient. We considered µ’=0.2 and 0.5, the boundaries of its reasonable range determined from focal mechanisms in Taiwan. Considering the stress threshold of ∆CFS≥0.1 bar and a distance threshold of 5 km, the potential paired structures were identified (Table 6). The results suggest slight differences within the reasonable effective friction coefficient (lines 5456, 259267). Besides, we explained our model without implementing a poroelastic assumption since previous studies (e.g., Chan and Stain, 2009) concluded that the differences in their results were trivial for assuming reasonable values of Skempton’s coefficients (between 0.5 and 0.9) and dry friction (0.75) (lines 259262).
minor comments:
Line 131, show in equation 1 > equation 3
This sentence has been removed.
Line 157, what is characteristic earthquake means? rupture or slip or magnitude?
This paragraph has been removed.
Line 159~ , The author addresses the exact value of each parameter very carefully, but I do think those repetitive equations and number should be removed and only use a simple table to present. Line 284, missing the ID for Chiayi frontal structure.
This paragraph has been removed.

AC8: 'The revised manuscript with annotation is available online', ChungHan Chan, 14 May 2022
The revised manuscript with annotation is available online in the following link:
https://editor.copernicus.org/index.php?_mdl=msover_md&_jrl=7&_lcm=oc108lcm109w&_acm=get_comm_sup_file&_ms=101260&c=224615&salt=4901848291507617197
or at AC5: 'Annotated manuscript'

AC8: 'The revised manuscript with annotation is available online', ChungHan Chan, 14 May 2022

AC3: 'Response to Reviewer #3 Anonymous Referee', ChungHan Chan, 14 May 2022

RC4: 'Comment on nhess202246', Alexandra Carvalho, 21 Mar 2022
There is a need that the fault ruptures complexities be accurately represented in hazard models, as in most PSHA studies the multiplefault rupture is neglected.
Being so, this manuscript addresses an updated problem and a discussion theme that should be debated among the scientific community.
Nevertheless, theories or new approaches and assumptions should rely on physical processes and need to incorporate the comprehension of the reality and I have some doubts about validity of some assumptions, namely:
 what is the meaning of summing slip rates of the different faults?
 Why the distance between two faults must be less than 5 km? is there any evidence that there is no Coulomb stress transfer for distances greater than 5 km that can trigger a fault? I could recommend a little more discussion on this issue.
The title “Quantifying the probability and uncertainty ….” do not reflect the content of the paper, in my opinion, as it leads to an expectation of a sensitivity study on key parameters that might have impact on results. The only parameters changed were the Coulomb stress and the structure rake angle. Are there any other parameters that can affect results? Were these parameters chosen because they are the ones with the most impact? I was expecting a more exhaustive study on that.
Finally, I would suggest a different way to present so many and so similar equations, as it become difficult and not very interesting to follow.

AC4: 'Response to Reviewer #4 Alexandra Carvalho', ChungHan Chan, 14 May 2022
We highly appreciate the reviewer’s insightful comments and have revised our manuscript, nhess202246, entitled, “Quantifying the probability and uncertainty of multiplestructure rupture and recurrence intervals in Taiwan,” accordingly. Below, we have quoted the comments in italics and provided our detailed responses. All the changes are underlined in the revised manuscript.
Nevertheless, theories or new approaches and assumptions should rely on physical processes and need to incorporate the comprehension of the reality and I have some doubts about validity of some assumptions, namely:
 what is the meaning of summing slip rates of the different faults?
The estimate of the multiplerupture slip rate through a sum is based on the assumption that the slip of an earthquake is equal to the cumulative slip during an interseismic period. Since the slip of a multiplestructure rupture is the result of contributions from different structures, we sum the slip rates contributed from the individual structures.
To better describe our algorithm for evaluating the recurrence interval of multiplestructure ruptures, we have adjusted the manuscript to first introduce the slip rate partitioned to individual structure ruptures (equations 8 and 9), followed by the obtained partitioned rates (equations 10 and 11). By combining them, the slip rate partitioned to the multiplestructure rupture from the original structures could be obtained (described in lines 123140).
 Why the distance between two faults must be less than 5 km? is there any evidence that there is no Coulomb stress transfer for distances greater than 5 km that can trigger a fault? I could recommend a little more discussion on this issue.
We expected that a long distance between two structures could result in it being difficul for the pair to rupture simultaneously. Thus, we followed the criterion by the UCERF3 (Field et al., 2015) and assumed a distance threshold of 5 km.
We were aware that an earthquake with a large coseismic slip dislocation could result in a significant stress change at distance and then searched the pairs with longer distances and significant stress increase. Two additional distance thresholds of 10 and 20 km were considered (Table 7). Generally, potential magnitudes of these structures are relatively large, which could result in leger stress perturbation.
The title “Quantifying the probability and uncertainty ….” do not reflect the content of the paper, in my opinion, as it leads to an expectation of a sensitivity study on key parameters that might have impact on results. The only parameters changed were the Coulomb stress and the structure rake angle. Are there any other parameters that can affect results? Were these parameters chosen because they are the ones with the most impact? I was expecting a more exhaustive study on that.
We followed the reviewer’s comment and included additional discussion on uncertainties from various parameters, including the effective coefficient of friction (Table 6, lines 259267), rake angle rotation (Table 8, lines 282285), stress threshold of ∆CFS (Table 3, lines 8994, 268272), and distance threshold (Tables 3 and 7, lines 8994, 268272, 273281). Our approach indicated various rupture pairs and quantified uncertainties. These outcomes were able to be incorporated into a probabilistic seismic hazard assessment through a logic tree.
Finally, I would suggest a different way to present so many and so similar equations, as it become difficult and not very interesting to follow.
We have rearranged the description of the procedure, simplified some equations, and removed several examples in Chapter 3. Hopefully, the current manuscript is easily understandable and meets the standards of Natural Hazards and Earth System Sciences.

AC9: 'The revised manuscript with annotation is available online', ChungHan Chan, 14 May 2022
The revised manuscript with annotation is available online in the following link:
https://editor.copernicus.org/index.php?_mdl=msover_md&_jrl=7&_lcm=oc108lcm109w&_acm=get_comm_sup_file&_ms=101260&c=224615&salt=4901848291507617197
or at AC5: 'Annotated manuscript'

AC9: 'The revised manuscript with annotation is available online', ChungHan Chan, 14 May 2022

AC4: 'Response to Reviewer #4 Alexandra Carvalho', ChungHan Chan, 14 May 2022

AC5: 'Annotated manuscript', ChungHan Chan, 14 May 2022
The comment was uploaded in the form of a supplement: https://nhess.copernicus.org/preprints/nhess202246/nhess202246AC5supplement.pdf
Status: closed

RC1: 'Comment on nhess202246', Jack Williams, 27 Feb 2022

AC1: 'Response to Reviewer #1 Jack Williams', ChungHan Chan, 14 May 2022
We greatly appreciate the reviewer’s insightful comments and have revised our manuscript, nhess202246, entitled, “Quantifying the probability and uncertainty of multiplestructure rupture and recurrence intervals in Taiwan,” accordingly. Below, we have quoted the comments in italics and provided our detailed responses. All the changes are underlined in the revised manuscript.
1.) Description of model
The key innovation of this study is described in Section 3.1 where it is outlined how the area and slip rate (i.e., a moment rate) of two different seismogenic structures can be combined with a GR relationship to determine the recurrence interval of an earthquake that ruptures both structures. As far as I can tell, there is nothing inherently wrong with the approach itself, however, I have several recommendations for how the presentation of this model could be improved.
We appreciate the reviewer’s very helpful recommendations. We considered the reviewer’s comments and responded in the following.
Immediately after equations 8 and 9 (Line 129), the meaning 𝐷̇𝐿1 (original L1 slip rate measurement) is given, but it is 𝐷̇𝐿1′ (slip rate for L1 single structure events) that is used in these equations. I would also present the equations for C1 (partitioning coefficient between 𝐷̇𝐿1+𝐿2𝐿1 and 𝐷̇𝐿1′, currently eqs. 10 and 11) and 𝐷̇𝐿1′ (currently eqs. 12 and 13) before the equation for 𝐷̇𝐿1+𝐿2𝐿1 (slip rate of L1 in L1+L2 events) given that you need these parameters to calculate 𝐷̇𝐿1+𝐿2𝐿1.
To clearly describe our algorithm for evaluating recurrence interval of multiplestructure ruptures, we first introduced the slip rate partitioned to individual structure ruptures (equations 8 and 9), followed by the obtained partitioned rates (equations 10 and 11). By combining them, the slip rate partitioned to the multiplestructure rupture from the original structures can be obtained (described in lines 123140).
I appreciate that the authors use the Hsinhua and Houchiali faults to provide an example of how their workflow is applied. However, showing the application of each equation to each structure in the text can get repetitive. I would suggest using a table to illustrate these equations, with a column for each structure. Another table could also be used for description of the model where >2 structures are considered (i.e., the example of the Chiayi, Meishan, and Tainan structures in Section 3.2).
In the previous manuscript, this example is provided to demonstrate the procedure of the workflow. To simplify the description of the calculation, this example has been removed.
When using the examples from the TEM, the values are provided to a high, and probably unjustified level of specificity (e.g. slip rates to 0.01 mm/yr, source areas to 0.01 km2, recurrence intervals to 1 year). I suggest rounding these values to a level appropriate with the uncertainty of this analysis.
We followed the reviewer’s comment and revised Table 1 accordingly. Now the slip rate and slip area are rounded to one decimal place and the nearest whole number, respectively. Note that we keep recurrence intervals to 1 year, since some structures (e.g., the Milun fault) obtain short recurrence intervals (<100 years).
Finally, the Wells and Coppersmith (1994) scaling relationships are increasingly out of date given that we now have nearly 30 more years’ worth of observed earthquakes to refine these relationships. I would recommend that either a more up to date set of scaling relationships is used (e.g., Leonard 2010, Thingbaijam et al 2017), or a sensitivity analysis is made to see if using the updated scaling relationships changes the model outcomes.
To validate the sensitivity of our procedure to scaling, we also implemented alternative relationships proposed by Yen and Ma (2011), who investigated the rupture parameters of the earthquakes mainly from the Taiwan orogenic belt.
Based on this relation, recurrence intervals for each multiplestructure rupture pairs were evaluated (Table 5). Comparing these with those obtained by Wells and Coppersmith’s relations, shorter recurrence intervals were obtained, especially for those with larger magnitude. These results can be attributed to a smaller average displacement obtained for a large event that led to a shorter recurrence interval for the multiplestructure rupture (based on equation 17). Note that although the scaling relations proposed by Wells and Coppersmith (1994) have been questioned by many modern models, especially for large megathrusts, Wang et al. (2016^{b}) concluded similar maximal magnitude of each seismogenic structure estimated from the relations of Wells and Coppersmith (1994) and Yen and Ma (2011). We provided more detailed descriptions in lines 214223, 292298.
Multistructure earthquakes are considered here only in terms of static Coulomb stress triggering between neighbouring faults. However, it is worth acknowledging in Section 4.2 that multistructure earthquakes may also be generated by dynamic stress triggering from seismic waves (e.g., Brodsky and van der Elst 2014, Ulrich et al 2018). I think this may what is being discussed at Line 280 (?), though note the reference is to a manuscript (Jiao et al 2020) that was not accepted for publication.
We followed the reviewer’s comment and indicated dynamic models could also constrain the behaviors of multiplestructure ruptures (lines 242245). Note that the paper by Jiao et al. has been published in 2022.
A key assumption in this study is that the magnitudefrequency distribution (MFD) of events along a single multistructure systems follow a GR scaling. Although that is certainly possible, one could also argue that at the scale of a single multistructure system, the MFD follows a characteristic shape (Youngs and Coppersmith 1984; Hecker et al 2013; Stirling and Zungia 2017), or that the MFD is neither characteristic nor GR (Geist and Parsons 2019; Page et al 2021). In either case, a deviation from a GR scaling will affect the recurrence intervals calculated through this model.
We are aware of the importance of the magnitudefrequency distribution (MFD) on a singlestructure rupture, and the MFD could be in various forms, including the GutenbergRichter law and the characteristic earthquake model. In this study, we evaluated the rupture recurrence interval as the ratio of slip of a characteristic earthquake (with maximum magnitude of the structure) and slip rate based on the assumption proposed by the TEM seismogenic structure database and the TEM PSHA2020. Note that this factor could be replaced by other magnitudefrequency distributions since the recurrence interval of the multiplestructure rupture in our procedure is based on slip rate partitioned from individual structure ruptures (shown as equations 89, 14, 18, and 20). We provided more detailed descriptions in lines 101104, 299307.
This model should also be discussed in the context of other studies that have attempted to incorporate multistructure ruptures in PSHA. For example, there are many studies that divide mapped multistructure systems into smaller subfault scale segments, and then essentially allow ruptures to ‘float’ across theses smaller segments in such a way that they fit a regional MFD target (Field et al 2014; 2021; Chartier et al 2019; Geist and Parsons 2019). These studies are therefore distinct from the model described here, which is quite prescriptive about the number of configurations that structures in the TEM can rupture in (i.e., as single or multistructure events only, and no events may be smaller than a single structure). It would benefit this study if the pros and cons of these different techniques could be discussed in Section 4.3.
Based on the assumption of the TEM PSHA2020, every rupture on a seismogenic structure results in a characteristic earthquake (with maximum magnitude of the structure), that is, small earthquakes (with magnitude smaller than the maximum magnitude of the structure) are attributed to shallow background sources. Following this assumption, we did not consider ruptures on small segments of a structure.
Minor Comments
Lines 721: The abstract does not mention that this study is using faults incorporated into the Taiwan Earthquake Model to perform this analysis. Suggest revise, Line 11 could be revised to:
‘……the probability of Coulomb stress triggering between seismogenic structures included in the Taiwan Earthquake Model.’
We followed the reviewer’s comment and revised the text accordingly.
Lines 6468: What value is used for the effective coefficient of friction (μ‘) in the Coulomb stress modelling?
We first assumed a fixed μ‘ of 0.4. To quantify deviation on determining multiplerupture pairs, we further considered µ’=0.2 and 0.5, the boundaries of its reasonable range determined from focal mechanisms in Taiwan. Considering the stress threshold of ∆CFS≥0.1 bar and a distance threshold of 5 km, the potential paired structures were identified (Table 6). The results suggest slight differences within the reasonable effective friction coefficient (lines 5456, 259267).
Line 115: These scaling relationships between magnitude and rupture area are presumably from Wells and Coppersmith (1994)? If so, they should be cited as such (though also see major comment #1)
We followed the reviewer’s comment and cited the reference accordingly.
Line 144: Replace ‘integrating’ with ‘combining,’ to avoid any connotations that you are actually performing an integration in these equations.
We followed the reviewer’s comment and revised the text accordingly.
Lines 220221 (and 335): When referring to the Kaikōura earthquake, reference should be made to Hamling et al (2017). This is the original reference to this event and written by authors who made the primary observations of this multifault earthquake.
We followed the reviewer’s comment and cited the reference accordingly.
Line 258: I think there is a typo here for describing the numeric value if the Hukou and Hsinchu fault recurrence intervals as ‘4.4 and 5.3’?
We have revised the text as “their recurrence intervals become 4.4 and 5.3 times, respectively, longer than the cases without considering multiplestructure ruptures” (lines 212213).
Figures: Figure 1 presents only a generic case of Coulomb stress changes around a fault. I would recommend also including a figure to show an example of this stress modelling from faults in the TEM. Maybe using the example of faults that are described further in Section 3.1?
This figure has been removed.
 AC6: 'The annotated manuscript is available', ChungHan Chan, 14 May 2022

AC11: 'The revised manuscript with annotation is available online', ChungHan Chan, 14 May 2022
The revised manuscript with annotation is available online in the following link:
https://editor.copernicus.org/index.php?_mdl=msover_md&_jrl=7&_lcm=oc108lcm109w&_acm=get_comm_sup_file&_ms=101260&c=224615&salt=4901848291507617197
or at AC5: 'Annotated manuscript'

AC1: 'Response to Reviewer #1 Jack Williams', ChungHan Chan, 14 May 2022

RC2: 'Comment on nhess202246', João Fonseca, 03 Mar 2022
Chang et al.´s manuscript address an interesting and relevant problem: three decades after the observation that two separate faults can interact in such ways that the rupture of one may trigger the rupture of the other (Stein et al., 1992), how can we incorporate this phenomenon in hazard assessment? The authors use a database of seismogenic faults in Taiwan to investigate the likelihood that such stress triggering of earthquakes could take place over a given period of time, and to quantify its impact on seismic hazard.
The authors apply an analytical procedure inspired in Chan et al. (2020), which stems from the notion that when two faults interact each one gives part of its slip rate to the multiplerupture process, while retaining the remaining slip rate to its individualrupture process. Starting from this basic notion – the “partitioned slip rates” of Chan et al. (2020)  the authors use Kanamori’s (1977) definition of moment magnitude, the definition of seismic moment, Wells and Coppersmith’s (1994) scaling relations of moment magnitude with rupture area and the GutenbergRichter relation to obtain return periods for multiple ruptures and for individual ruptures on each fault.
Key to the authors' reasoning is their assumption that the slip rate of each fault is partitioned between individual and multiple ruptures according to a “partitioned rate” given by their equations 10 and 11, which include "the magnitude of the multiplestructure rupture” (line 135) and “the displacement of the multiplerupture structure” (line 136). This terminology reflects the fact (not explained in the manuscript) that the catalog used in the study considers characteristic ruptures only. The expression for C ifeatures the bvalue of the GutenbergRichter relation. The authors present the expression for C as a logical conclusion of the GutenbergRichter relation, although I was not able to follow that logic. I was able to trace the definition of the partitioned rate C to Chan et al. (2020), but there too it was introduced without an explanation.
The authors estimate a return period for a multiple rupture in a pair of faults as R_{12}=D_{12}/D_{12}', where D_{12} is a displacement associated with the joint rupture of the two faults and D_{12}' is a slip rate associated with the joint rupture of the two faults. To obtain D_{12} the authors use the definition of seismic moment inserting for A the sum of the two fault areas and estimating the seismic moment from a value of magnitude inferred from A using the Wells and Coppersmith (1994) scaling relations. To obtain the multiplerupture sliprate the authors assume that D_{12}'=D^{12}_{1}'+D^{12}_{2}', where D^{12}_{k}' is the part of the slip rate of faullt k that takes place through joint rupture (estimated with the partitioned rates discussed above). It is important to inquire into the physical meaning of these quantities. The estimate of the multiplerupture displacement is formally correct, albeit highly convoluted. But the estimate of the multiplerupture slip rate through a sum defies logic, in my view (why sum slip rates interesting separate faults?) Also, as pointed out above, each parcel relies on a coefficient that was not sufficiently explained.
In section 3.2 the authors enlarge their approach to include more than two faults in interaction, increasing the complexity while inheriting the obscurity from the previous section.
In sections 3.3 and 4, the authors discuss some implications of their analysis for seismic hazard. Around line 245, the authors conclude that the possibility of multiplerupture earthquakes reduces the hazard at the shorter return periods while increasing it at longer return periods. In line 255, the authors observe that “structures that pair with several cases of multiplestructure ruptures might be difficult to rupture solely”. These observations are so clearly at odds with empirical evidence – which points to singlefault rupture as the dominant contributor to hazard – that they should be regarded as indicating flaws of the approach.
The authors base their approach on a simplified view of stress transfer between faults: they ignore dynamic effects, porefluid effects and – surprisingly in view of published evidence – restrict the range of stress transfer to 5km. Although the title promised a quantification of the uncertainties, very little is done to quantify the errors that derive from such simplifications. In line 263 the authors state that their approach is a physicsbased one. Unfortunatelly, it seems to have strayed strongly from the geological reality of earthquake generation. The authors recognize, to their credit, that the “analysis could be further improved through better understanding seismogenic structures” (line 278). I would take this conclusion even further and say that the analysis needs to be reformulated starting with a better understanding of seismogenic processes. For example, exploring empirical evidence of the occurrence and characteristics of multiplerupture earthquakes in the available databases, in order to be able to subject their model to a reality check.
In the present stage of development, I regret to conclude that I don’t consider this research ready for publication.
Lisbon, March 3, 2022
Joao Fonseca

AC2: 'Response to Reviewer #2 João Fonseca', ChungHan Chan, 14 May 2022
We greatly appreciate the reviewer’s insightful comments and have revised our manuscript, nhess202246, entitled, “Quantifying the probability and uncertainty of multiplestructure rupture and recurrence intervals in Taiwan,” accordingly. Below, we have quoted the comments in italics and provided our detailed responses. All the changes are underlined in the revised manuscript.
The authors apply an analytical procedure inspired in Chan et al. (2020), which stems from the notion that when two faults interact each one gives part of its slip rate to the multiplerupture process, while retaining the remaining slip rate to its individualrupture process. Starting from this basic notion – the “partitioned slip rates” of Chan et al. (2020)  the authors use Kanamori’s (1977) definition of moment magnitude, the definition of seismic moment, Wells and Coppersmith’s (1994) scaling relations of moment magnitude with rupture area and the GutenbergRichter relation to obtain return periods for multiple ruptures and for individual ruptures on each fault. Key to the authors' reasoning is their assumption that the slip rate of each fault is partitioned between individual and multiple ruptures according to a “partitioned rate” given by their equations 10 and 11, which include "the magnitude of the multiplestructure rupture” (line 135) and “the displacement of the multiplerupture structure” (line 136). This terminology reflects the fact (not explained in the manuscript) that the catalog used in the study considers characteristic ruptures only. The expression for C ifeatures the bvalue of the GutenbergRichter relation. The authors present the expression for C as a logical conclusion of the GutenbergRichter relation, although I was not able to follow that logic. I was able to trace the definition of the partitioned rate C to Chan et al. (2020), but there too it was introduced without an explanation.
I replied this comment through three aspects, algorithm description, scaling relation, and assumption of characteristic ruptures, detailed below.
To clearly describe our algorithm for evaluating the recurrence interval of multiplestructure ruptures, we first introduced the slip rate partitioned to individual structure ruptures (equations 8 and 9), followed by the obtained partitioned rates (equations 10 and 11). By combining them, the slip rate partitioned to the multiplestructure rupture from the original structures could be obtained (described in lines 122140).
Although the scaling relations proposed by Wells and Coppersmith (1994) have been questioned by many modern models, especially for large megathrusts, Wang et al. (2016^{b}) concluded a similar maximal magnitude of each seismogenic structure estimated from the relations of Wells and Coppersmith (1994) and Yen and Ma (2011), obtained from regressions of the rupture parameters of the earthquakes mainly from the Taiwan orogenic belt. Besides, to validate the sensitivity of our procedure to scaling, we implemented alternative relationships proposed by Yen and Ma (2011). Based on this relation, recurrence intervals for each multiplestructure rupture pairs were evaluated (Table 5). Comparing these with those obtained by Wells and Coppersmith’s relations, shorter recurrence intervals were obtained, especially for those with larger magnitude. These results can be attributed to a smaller average displacement obtained for a large event that led to a shorter recurrence interval for the multiplestructure rupture (based on equation 17). We provided more detailed descriptions in lines 214223, 292298.
The estimate of the multiplerupture displacement is formally correct, albeit highly convoluted. But the estimate of the multiplerupture slip rate through a sum defies logic, in my view (why sum slip rates interesting separate faults?) Also, as pointed out above, each parcel relies on a coefficient that was not sufficiently explained.
To clearly describe our algorithm for evaluating the recurrence interval of multiplestructure ruptures, we have modified the manuscript to first introduce the slip rate partitioned to individual structure ruptures (equations 8 and 9), followed by the obtained partitioned rates (equations 10 and 11). By combining them, the slip rate partitioned to the multiplestructure rupture from the original structures can be obtained (described in lines 122140).
The estimate of the multiplerupture slip rate through a sum is based on the assumption that the slip of an earthquake is equal to the cumulative slip during an interseismic period. Since the slip of a multiplestructure rupture is the result of contributions from different structures, we sum the slip rates contributed from the individual structures.
In section 3.2 the authors enlarge their approach to include more than two faults in interaction, increasing the complexity while inheriting the obscurity from the previous section.
Earthquakes could be attributed to multiple (more than three) structures, for example, the 2010 El MayorCucapah, US, earthquake; the 2016 Mw7.8 Kaikōura, New Zealand, earthquake. The procedure we proposed in Section 3.2 could quantify the return period of these earthquakes.
In sections 3.3 and 4, the authors discuss some implications of their analysis for seismic hazard. Around line 245, the authors conclude that the possibility of multiplerupture earthquakes reduces the hazard at the shorter return periods while increasing it at longer return periods. In line 255, the authors observe that “structures that pair with several cases of multiplestructure ruptures might be difficult to rupture solely”. These observations are so clearly at odds with empirical evidence – which points to singlefault rupture as the dominant contributor to hazard – that they should be regarded as indicating flaws of the approach.
The description mentioned here is based on the comparison between models with and without multiplestructure ruptures. That is, the return period of a seismogenic structure could be longer if a part of its coupling rate will contribute to the multiplestructure rupture. Note that based on our procedure, a shorter return period is expected for a rupture on one individual structure than for a multiplestructure rupture. For example, we obtained a return period of 6,640 and 11,953 years for the Hsinchu fault and multiplestructure rupture of the Hukou fault and Hsinchu fault, respectively.
The authors base their approach on a simplified view of stress transfer between faults: they ignore dynamic effects, porefluid effects and – surprisingly in view of published evidence – restrict the range of stress transfer to 5km. Although the title promised a quantification of the uncertainties, very little is done to quantify the errors that derive from such simplifications. In line 263 the authors state that their approach is a physicsbased one. Unfortunatelly, it seems to have strayed strongly from the geological reality of earthquake generation. The authors recognize, to their credit, that the “analysis could be further improved through better understanding seismogenic structures” (line 278). I would take this conclusion even further and say that the analysis needs to be reformulated starting with a better understanding of seismogenic processes. For example, exploring empirical evidence of the occurrence and characteristics of multiplerupture earthquakes in the available databases, in order to be able to subject their model to a reality check.
We followed the reviewer’s comment and included some more discussion on various physicsbased components, including effective coefficient of friction (Table 6, lines 259267), rake angle rotation (Table 8, lines 282285), stress threshold of ∆CFS (Table 3, lines 8994, 268272), and distance threshold (Tables 3 and 7, lines 8994, 268272, 273281). Note that we explained our model without implementing a poroelastic assumption, since previous studies (e.g., Chan and Stain, 2009) concluded that the differences in their results were trivial for assuming reasonable values of Skempton’s coefficients (in between 0.5 and 0.9) and dry friction (0.75). Our approach indicated various rupture pairs and quantified uncertainties. These outcomes could be incorporated into a probabilistic seismic hazard assessment through a logic tree.
In the present stage of development, I regret to conclude that I don’t consider this research ready for publication.
We appreciate the reviewer’s very helpful comments. We hope the adjustments we have made accordingly to the manuscript meet the standards of Natural Hazards and Earth System Sciences and have made the manuscript to now ready for publication.
 AC7: 'The annotated manuscript is available', ChungHan Chan, 14 May 2022

AC10: 'The revised manuscript with annotation is available online', ChungHan Chan, 14 May 2022
The revised manuscript with annotation is available online in the following link:
https://editor.copernicus.org/index.php?_mdl=msover_md&_jrl=7&_lcm=oc108lcm109w&_acm=get_comm_sup_file&_ms=101260&c=224615&salt=4901848291507617197
or at AC5: 'Annotated manuscript'

AC2: 'Response to Reviewer #2 João Fonseca', ChungHan Chan, 14 May 2022

RC3: 'Comment on nhess202246', Anonymous Referee #3, 05 Mar 2022
Huang et al., in the manuscript "Quantifying the probability and uncertainty of multiplestructure rupture and recurrence intervals in Taiwan" presents a new approach by integrating the physicsbased model (static Coulomb stress change) and statistic model (GutenbergRichter law) to evaluate the earthquake recurrence time for the possible multiplerupture scenario. According to their assumption, multirupture only occurs if the stress transfer on the nearby fault reaches a certain value, and the slip rate of the multiplerupture structure is the sum of the associated slip rate in related ruptures. Although I acknowledge this topic as a valuable contribution in the field of hazard assessment, however, this current manuscript needs improvements, especially I am still not clear about how the author partitioned the slip rates between different ruptures.
The structure of the description.
I think section 3 is the core of the methodology in this study, as far as I can tell this study use simple equations, but the description makes it extremely difficult to follow.In general, I think the whole section of 3.1 and 3.2 should be reformulate, for example:
Equation (2),D^dot represents the slip rate, dose this slip rate indicates the longterm slip rate obtained from other measurements?Equation (7), the author used the MwMo scaling law by Kanamori (1977), but the equation in the manuscript is from Hanks and Kanamori (1979) with the unit of dynecm.
Equation (8) and (9), there appear two parameters D_L1’ and D_L2’ with no explanations until equation (12) and equation (13).
Equation (10), dose the ML1 indicates the maximum magnitude in L1 ?
D_L1+L2 is the displacement of the multiplestructure rupture, dose this means D_L1+L2 = D_L1 + D_L2? More practical parameter annotation should be carefully addressed.Equation (14), this equation is hard to follow, in Line 146 : the sum of the slip rates for the multiplestructure…. I don’t understand what is the sum of the slip rates for the multiplestructure? and this statement is not correspond to the equation (14).
The discussion:
The author took 1906 Meishan earthquake as an example, they argued that closedby Chiayi frontal structure also ruptured during the coseismic period because liquefaction took place on the west of the Meishan fault, however, I think this statement is littlebit weak because liquefaction could occur when the stress is perturbated through seismic wave propagation from the mainshock. Also, I got confused when reading the line from 286 to 288, dose the author really hints that Meishan earthquake is initiated on the Chiayi frontal structure?For model uncertainty, this sensitivity test is focus only on the rake angles for estimating the Coulomb stress change, I was wondering what if they change the friction coefficient? Friction coefficient also plays an important role on evaluating the stress impart from the mainshock, especially recent studies suggest that friction coefficient is depth dependence (i.e., Carpenter et al., 2012,2015). Besides the Coulomb stress model, GR law also make a strong contribution on this approach, I am wondering if they consider different type of GR law will change the result significantly (for example the truncated model)?
minor comments:
Line 131, show in equation 1 > equation 3
Line 157, what is characteristic earthquake means? rupture or slip or magnitude?
Line 159~ , The author addresses the exact value of each parameter very carefully, but I do think those repetitive equations and number should be removed and only use a simple table to present.
Line 284, missing the ID for Chiayi frontal structure

AC3: 'Response to Reviewer #3 Anonymous Referee', ChungHan Chan, 14 May 2022
We highly appreciate the reviewer’s insightful comments and have revised our manuscript, nhess202246, entitled, “Quantifying the probability and uncertainty of multiplestructure rupture and recurrence intervals in Taiwan,” accordingly. Below, we have quoted the comments in italics and provided our detailed responses. All the changes are underlined in the revised manuscript.
Huang et al., in the manuscript "Quantifying the probability and uncertainty of multiplestructure rupture and recurrence intervals in Taiwan" presents a new approach by integrating the physicsbased model (static Coulomb stress change) and statistic model (GutenbergRichter law) to evaluate the earthquake recurrence time for the possible multiplerupture scenario. According to their assumption, multirupture only occurs if the stress transfer on the nearby fault reaches a certain value, and the slip rate of the multiplerupture structure is the sum of the associated slip rate in related ruptures. Although I acknowledge this topic as a valuable contribution in the field of hazard assessment, however, this current manuscript needs improvements, especially I am still not clear about how the author partitioned the slip rates between different ruptures.
To clearly describe our algorithm for slip rate partitioning, we revised our procedure to first introduce the slip rate partitioned to individual structure ruptures (equations 8 and 9), followed by the obtained partitioned rates (equations 10 and 11). By combining them, the slip rate partitioned to the multiplestructure rupture from the original structures could be obtained (described in Lines 123140).
We hope the present version of the manuscript meets the standards of Natural Hazards and Earth System Sciences and is now ready for publication.
The structure of the description.
I think section 3 is the core of the methodology in this study, as far as I can tell this study use simple equations, but the description makes it extremely difficult to follow. In general, I think the whole section of 3.1 and 3.2 should be reformulate, for example:
Equation (2),D^dot represents the slip rate, dose this slip rate indicates the longterm slip rate obtained from other measurements?
The slip rate (_{L1}, shown in equation 2) is obtained from the TEM seismogenic structure database (Table 1).
To clearly describe our algorithm for the recurrence interval of multiplestructure ruptures, especially for slip rate partitioning, we modified Section 3 and hope the current version achieves the desired clarity (lines 97190).
Equation (7), the author used the MwMo scaling law by Kanamori (1977), but the equation in the manuscript is from Hanks and Kanamori (1979) with the unit of dynecm.
We thank the reviewer highly for having identified this oversight in our paper. We have revised the manuscript accordingly and simplified equation 7.
Equation (8) and (9), there appear two parameters D_L1’ and D_L2’ with no explanations until equation (12) and equation (13).
Equation (10), dose the ML1 indicates the maximum magnitude in L1 ? D_L1+L2 is the displacement of the multiplestructure rupture, dose this means D_L1+L2 = D_L1 + D_L2? More practical parameter annotation should be carefully addressed.
Equation (14), this equation is hard to follow, in Line 146 : the sum of the slip rates for the multiplestructure…. I don’t understand what is the sum of the slip rates for the multiplestructure? and this statement is not correspond to the equation (14).
To clearly describe our algorithm for evaluating the recurrence interval of multiplestructure ruptures, we first introduced the slip rate partitioned to individual structure ruptures (equations 8 and 9), followed by the obtained partitioned rates (equations 10 and 11). By combining them, the slip rate partitioned to the multiplestructure rupture from the original structures could be obtained (described in lines 123140).
The author took 1906 Meishan earthquake as an example, they argued that closedby Chiayi frontal structure also ruptured during the coseismic period because liquefaction took place on the west of the Meishan fault, however, I think this statement is littlebit weak because liquefaction could occur when the stress is perturbated through seismic wave propagation from the mainshock.
To better illustrate the rupture behavior of the Maishan earthquake, we provided evidence such as the larger magnitude than the characteristic magnitude of the Meishan fault, the focal mechanism of oblique thrust faulting being oriented in the northeast–southwest direction, and the large ground shaking with liquefaction that took place to the west. All infer the Chiayi frontal structure might rupture simultaneously.
Also, I got confused when reading the line from 286 to 288, dose the author really hints that Meishan earthquake is initiated on the Chiayi frontal structure?
The description of the simplified Coulomb stress change model has been removed.
For model uncertainty, this sensitivity test is focus only on the rake angles for estimating the Coulomb stress change, I was wondering what if they change the friction coefficient? Friction coefficient also plays an important role on evaluating the stress impart from the mainshock, especially recent studies suggest that friction coefficient is depth dependence (i.e., Carpenter et al., 2012,2015). Besides the Coulomb stress model, GR law also make a strong contribution on this approach, I am wondering if they consider different type of GR law will change the result significantly (for example the truncated model)?
We followed the reviewer’s comment and discussed the impact of the friction coefficient. We considered µ’=0.2 and 0.5, the boundaries of its reasonable range determined from focal mechanisms in Taiwan. Considering the stress threshold of ∆CFS≥0.1 bar and a distance threshold of 5 km, the potential paired structures were identified (Table 6). The results suggest slight differences within the reasonable effective friction coefficient (lines 5456, 259267). Besides, we explained our model without implementing a poroelastic assumption since previous studies (e.g., Chan and Stain, 2009) concluded that the differences in their results were trivial for assuming reasonable values of Skempton’s coefficients (between 0.5 and 0.9) and dry friction (0.75) (lines 259262).
minor comments:
Line 131, show in equation 1 > equation 3
This sentence has been removed.
Line 157, what is characteristic earthquake means? rupture or slip or magnitude?
This paragraph has been removed.
Line 159~ , The author addresses the exact value of each parameter very carefully, but I do think those repetitive equations and number should be removed and only use a simple table to present. Line 284, missing the ID for Chiayi frontal structure.
This paragraph has been removed.

AC8: 'The revised manuscript with annotation is available online', ChungHan Chan, 14 May 2022
The revised manuscript with annotation is available online in the following link:
https://editor.copernicus.org/index.php?_mdl=msover_md&_jrl=7&_lcm=oc108lcm109w&_acm=get_comm_sup_file&_ms=101260&c=224615&salt=4901848291507617197
or at AC5: 'Annotated manuscript'

AC8: 'The revised manuscript with annotation is available online', ChungHan Chan, 14 May 2022

AC3: 'Response to Reviewer #3 Anonymous Referee', ChungHan Chan, 14 May 2022

RC4: 'Comment on nhess202246', Alexandra Carvalho, 21 Mar 2022
There is a need that the fault ruptures complexities be accurately represented in hazard models, as in most PSHA studies the multiplefault rupture is neglected.
Being so, this manuscript addresses an updated problem and a discussion theme that should be debated among the scientific community.
Nevertheless, theories or new approaches and assumptions should rely on physical processes and need to incorporate the comprehension of the reality and I have some doubts about validity of some assumptions, namely:
 what is the meaning of summing slip rates of the different faults?
 Why the distance between two faults must be less than 5 km? is there any evidence that there is no Coulomb stress transfer for distances greater than 5 km that can trigger a fault? I could recommend a little more discussion on this issue.
The title “Quantifying the probability and uncertainty ….” do not reflect the content of the paper, in my opinion, as it leads to an expectation of a sensitivity study on key parameters that might have impact on results. The only parameters changed were the Coulomb stress and the structure rake angle. Are there any other parameters that can affect results? Were these parameters chosen because they are the ones with the most impact? I was expecting a more exhaustive study on that.
Finally, I would suggest a different way to present so many and so similar equations, as it become difficult and not very interesting to follow.

AC4: 'Response to Reviewer #4 Alexandra Carvalho', ChungHan Chan, 14 May 2022
We highly appreciate the reviewer’s insightful comments and have revised our manuscript, nhess202246, entitled, “Quantifying the probability and uncertainty of multiplestructure rupture and recurrence intervals in Taiwan,” accordingly. Below, we have quoted the comments in italics and provided our detailed responses. All the changes are underlined in the revised manuscript.
Nevertheless, theories or new approaches and assumptions should rely on physical processes and need to incorporate the comprehension of the reality and I have some doubts about validity of some assumptions, namely:
 what is the meaning of summing slip rates of the different faults?
The estimate of the multiplerupture slip rate through a sum is based on the assumption that the slip of an earthquake is equal to the cumulative slip during an interseismic period. Since the slip of a multiplestructure rupture is the result of contributions from different structures, we sum the slip rates contributed from the individual structures.
To better describe our algorithm for evaluating the recurrence interval of multiplestructure ruptures, we have adjusted the manuscript to first introduce the slip rate partitioned to individual structure ruptures (equations 8 and 9), followed by the obtained partitioned rates (equations 10 and 11). By combining them, the slip rate partitioned to the multiplestructure rupture from the original structures could be obtained (described in lines 123140).
 Why the distance between two faults must be less than 5 km? is there any evidence that there is no Coulomb stress transfer for distances greater than 5 km that can trigger a fault? I could recommend a little more discussion on this issue.
We expected that a long distance between two structures could result in it being difficul for the pair to rupture simultaneously. Thus, we followed the criterion by the UCERF3 (Field et al., 2015) and assumed a distance threshold of 5 km.
We were aware that an earthquake with a large coseismic slip dislocation could result in a significant stress change at distance and then searched the pairs with longer distances and significant stress increase. Two additional distance thresholds of 10 and 20 km were considered (Table 7). Generally, potential magnitudes of these structures are relatively large, which could result in leger stress perturbation.
The title “Quantifying the probability and uncertainty ….” do not reflect the content of the paper, in my opinion, as it leads to an expectation of a sensitivity study on key parameters that might have impact on results. The only parameters changed were the Coulomb stress and the structure rake angle. Are there any other parameters that can affect results? Were these parameters chosen because they are the ones with the most impact? I was expecting a more exhaustive study on that.
We followed the reviewer’s comment and included additional discussion on uncertainties from various parameters, including the effective coefficient of friction (Table 6, lines 259267), rake angle rotation (Table 8, lines 282285), stress threshold of ∆CFS (Table 3, lines 8994, 268272), and distance threshold (Tables 3 and 7, lines 8994, 268272, 273281). Our approach indicated various rupture pairs and quantified uncertainties. These outcomes were able to be incorporated into a probabilistic seismic hazard assessment through a logic tree.
Finally, I would suggest a different way to present so many and so similar equations, as it become difficult and not very interesting to follow.
We have rearranged the description of the procedure, simplified some equations, and removed several examples in Chapter 3. Hopefully, the current manuscript is easily understandable and meets the standards of Natural Hazards and Earth System Sciences.

AC9: 'The revised manuscript with annotation is available online', ChungHan Chan, 14 May 2022
The revised manuscript with annotation is available online in the following link:
https://editor.copernicus.org/index.php?_mdl=msover_md&_jrl=7&_lcm=oc108lcm109w&_acm=get_comm_sup_file&_ms=101260&c=224615&salt=4901848291507617197
or at AC5: 'Annotated manuscript'

AC9: 'The revised manuscript with annotation is available online', ChungHan Chan, 14 May 2022

AC4: 'Response to Reviewer #4 Alexandra Carvalho', ChungHan Chan, 14 May 2022

AC5: 'Annotated manuscript', ChungHan Chan, 14 May 2022
The comment was uploaded in the form of a supplement: https://nhess.copernicus.org/preprints/nhess202246/nhess202246AC5supplement.pdf
ChiehChen Chang et al.
ChiehChen Chang et al.
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