The Role of Disorder in Foreshock Activity

Round 1

Reviewer 1 Report

I think that this manuscript deals with important and relevant topics of the seismicity pattern models, but it seems to me too short and without quoting extensively the databases of seismicity taken into account in order to figure out the results described. Indeed, graphs and tables reporting the earthquakes sequences used for this study would have been helpful to the readers, in order to connect this study model to the real data , because the faults heterogeneity at depth is a very difficult parameter to be assessed quantitatively, in several tectonic environments.   

Author Response

The referee writes: I think that this manuscript deals with important and relevant topics of the seismicity pattern models, but it seems to me too short and without quoting extensively the databases of seismicity taken into account in order to figure out the results described. Indeed, graphs and tables reporting the earthquakes sequences used for this study would have been helpful to the readers, in order to connect this study model to the real data , because the faults heterogeneity at depth is a very difficult parameter to be assessed quantitatively, in several tectonic environments.    

Answer: I thank the referee for this constructive observation.
We would like to clarify that the aim of this study is not to model specific earthquake sequences from real catalogs, but to investigate, through a well-established numerical model, how fault heterogeneity affects the occurrence of foreshocks. 
The model employed here is designed to reproduce the universal statistical properties of seismicity - including the Gutenberg-Richter law, the Omori–Utsu law, productivity law, and spatial clustering. Therefore, no empirical earthquake catalog was used as an input for this study. Rather, our goal is to explore the model behavior under controlled variations of disorder, and to compare the trends qualitatively with laboratory experiments (Goebel et al., 2024) and previous theoretical work. 
In response to the referee’s valuable suggestion, and also following the recommendations of the editor and other referees, we have extended the manuscript to better highlight these concepts and to improve clarity for a broad readership.  

Reviewer 2 Report

This is a useful paper that examines the Role of Disorder in Foreshock Activity based on a numerical model of seismicity. The paper would be acceptable for publication after moderate revision according to the next recommendations.

In the introduction, it would be nice to briefly overview the two main physical models that have been proposed to explain the generation and evolution of foreshock sequences: the cascade model and the nucleation model.

22-23. A very good example that could be mentioned is the foreshock activity prior to the M6.8 earthquake in the Ionian Sea (see Papadopoulos et al., 2020).

Papadopoulos, G.A.; Agalos, A.; Minadakis, G.; Triantafyllou, I.; Krassakis, P. Short-Term Foreshocks as Key Information for Mainshock Timing and Rupture: The Mw6.8 25 October 2018 Zakynthos Earthquake, Hellenic Subduction Zone. Sensors 2020, 20, 5681. https://doi.org/10.3390/s20195681

43-44, 139, 144-145. Absolutely consistent are also the simulation results received in the paper by Im and Avouac (2024), which I recommend also including in the discussion and in the citations. 

Im, K.; Avouac, J.-Ph. Cascading foreshocks, aftershocks, and earthquake swarms in a discrete fault network. Geophys. J. Internat. 2023, 235, 831-852, https://doi.org/10.1093/gji/ggad278

131, 137, 164. Instead of Supplementary Information, you mean Appendix A1. Correct?

Appendix A1. In the text and in the main panel of Fig. A1, both curves show a decrease in foreshock percentage after a certain magnitude. Also, why don’t you extend the simulation to larger magnitudes? Because of the curve decrease? These issues need explanation in order to make your argument clear.

Author Response

The referee writes: This is a useful paper that examines the Role of Disorder in Foreshock Activity based on a numerical model of seismicity. The paper would be acceptable for publication after moderate revision according to the next recommendations. 
In the introduction, it would be nice to briefly overview the two main physical models that have been proposed to explain the generation and evolution of foreshock sequences: the cascade model and the nucleation model.
22-23. A very good example that could be mentioned is the foreshock activity prior to the M6.8 earthquake in the Ionian Sea (see Papadopoulos et al., 2020).
Papadopoulos, G.A.; Agalos, A.; Minadakis, G.; Triantafyllou, I.; Krassakis, P. Short-Term Foreshocks as Key Information for Mainshock Timing and Rupture: The Mw6.8 25 October 2018 Zakynthos Earthquake, Hellenic Subduction Zone. Sensors 2020, 20, 5681. https://doi.org/10.3390/s20195681 43-44, 139, 144-145. Absolutely consistent are also the simulation results received in the paper by Im and Avouac (2024), which I recommend also including in the discussion and in the citations. Im, K.; Avouac, J.-Ph. Cascading foreshocks, aftershocks, and earthquake swarms in a discrete fault network. Geophys. J. Internat. 2023, 235, 831-852, https://doi.org/10.1093/gji/ggad278 131, 137, 164.\\
Instead of Supplementary Information, you mean Appendix A1. Correct? 

Answer: I thank the referee for the positive evaluation of the manuscript and for the constructive suggestions to further improve it. Below we address each point in detail. 
We have included a citation and a brief mention of the Zakynthos 2018 earthquake as an example of real foreshock activity in the revised Discussion section. Also, we have added a citation and discussion of the work by Im \& Avouac (2023) in the Discussion section. 

The referee writes: Appendix A1. In the text and in the main panel of Fig. A1, both curves show a decrease in foreshock percentage after a certain magnitude. Also, why don’t you extend the simulation to larger magnitudes? Because of the curve decrease? These issues need explanation in order to make your argument clear. \\

Answer: We thank the referee for this important point. We have added an explicit explanation in the text of Appendix A1. In summary: 
The slight decrease in foreshock percentage at large mainshock magnitudes arises from "geometric saturation” effect, whereby very large events tend to nucleate more abruptly, leaving less opportunity for observable precursory activity. Extending the simulation to larger magnitudes is certainly feasible in this model: for example, by reducing the dissipation parameter (see Petrillo et al., 2020). However, this would not qualitatively alter the trends observed, while requiring significantly longer computational times due to the larger cascades. For this reason, we focused on a magnitude range where statistical reliability is good and the trends are robust. 
This clarification has been added to Appendix A1, as recommended.

Reviewer 3 Report

The manuscript titled "The Role of Disorder in Foreshock Activity" investigated the effect of disorder in foreshock patterns using mathematical-physical modeling. As most of my experience working on real-world seismic catalogs, I found the manuscript both interesting and difficult to understand, therefore, I raise some questions to clarify the research.

1. Please explain the physical meaning of the disorder parameter σ. What is the real-world equivalent of this parameter?
2. L90-94: Because the author uses the numerical model for this research, there would be no STAI; however, most of the real seismic catalog should have some incompleteness, and other tectonic-physical parameters affect the foreshock-mainshock-aftershock pattern. Could the author include these issues in the model?
3. L115-119 and Fig. 2 text, the author reports that the productivity exponent αf stabilizes around 1 and cites reference [22]. As I understand, the reference [22] is a modeling research, but in Fig. 2, the text "showing a good agreement with the experimental data", is there any reference from laboratory experiment or real-world data?
4. For both the results in Fig. 2 and Fig. 3, I found it difficult to understand the model. Could you provide the entire or some parts (main ideas) of the simulation code, or a step-by-step mathematical explanation, it would be much appreciated. 
5. Is there any difference in varying the model parameter θ? Does this parameter change result in any significant changes in the results? 
6. Although the author references some research that supports the model results, I suggest that the author should sum up the physical equivalent of the disorder parameter σ (similar to my question 1).
7. What is the contribution of this research to seismic hazard assessment (expand the ideas of L194-197)? How can seismologists quantify or apply this model to a real seismic sequence?

Small corrections:
Please make the reference format consistent in parts 4 and 5.

Author Response

The referee writes: The manuscript titled "The Role of Disorder in Foreshock Activity" investigated the effect of disorder in foreshock patterns using mathematical-physical modeling. As most of my experience working on real-world seismic catalogs, I found the manuscript both interesting and difficult to understand, therefore, I raise some questions to clarify the research. \\

Answer: I sincerely thank the referee for the careful reading of our manuscript and for finding the study interesting. Below we provide detailed responses to each of the referee’s questions and comments, aiming to clarify the methodology, assumptions, and relevance of the model results to real seismicity.  \\

The referee writes: 1. Please explain the physical meaning of the disorder parameter $\sigma$. What is the real-world equivalent of this parameter?    \\

Answer:  I thank the referee for this important question. The disorder parameter $\sigma$ in the model represents the variance of the distribution of local failure thresholds $f_i^{\text{th}}$, which physically correspond to the strength of individual fault patches or asperities. In real-world faults, this variability arises from multiple sources of heterogeneity, such as: variations in fault-zone lithology, differences in mineral composition, roughness of the fault surface, damage structures and gouge development, variations in local pore-fluid pressure, variations in frictional healing or fault maturity.
In this sense, $\sigma$ is a synthetic parameter that captures the combined effect of these sources of heterogeneity on the mechanical strength landscape of the fault. \\
I have now clarified this point in the text.\\

The referee writes: 2. L90-94: Because the author uses the numerical model for this research, there would be no STAI; however, most of the real seismic catalog should have some incompleteness, and other tectonic-physical parameters affect the foreshock-mainshock-aftershock pattern. Could the author include these issues in the model? \\

Answer:  I thank the referee for raising this important point. Indeed, in real seismic catalogs, Short-Term Aftershock Incompleteness (STAI) and other observational biases affect the recorded foreshock and aftershock patterns. In contrast, our numerical model provides an “idealized” catalog where all events are perfectly recorded, thus allowing us to isolate and analyze the effects of fault heterogeneity without the confounding influence of observational limitations. As for including such effects in the model, it is indeed possible and a promising direction for future work. In particular, STAI can be emulated by imposing a rate-dependent detection threshold after large events, following empirical formulations (Hainzl, 2016). In this first work, I chose to focus on the intrinsic physical effect of disorder in an idealized setting to clearly identify its role in foreshock activity. However, the model is fully compatible with the addition of observational filters or more realistic tectonic forcings, and we plan to explore such extensions in future studies. I included this point in the revised version of the manuscript. \\

The referee writes:3. L115-119 and Fig. 2 text, the author reports that the productivity exponent αf stabilizes around 1 and cites reference [22]. As I understand, the reference [22] is a modeling research, but in Fig. 2, the text "showing a good agreement with the experimental data", is there any reference from laboratory experiment or real-world data?  \\

Answer:  Thank you for this valuable observation. You are correct that reference Petrillo et al. 2020 is based on modeling. The phrase “showing a good agreement with the experimental data” was meant to refer to the fact that an exponent $\alpha_f \sim 1$ has also been reported in both laboratory experiments and real seismicity studies. For example, statistical analyses of earthquake catalogs, such as Lippiello et al. (2012), reports that the number of foreshocks grow exponentially with the mainshock magnitude, consistently with a productivity law.
To improve clarity, I revised the text. \\

The referee writes:4. For both the results in Fig. 2 and Fig. 3, I found it difficult to understand the model. Could you provide the entire or some parts (main ideas) of the simulation code, or a step-by-step mathematical explanation, it would be much appreciated.  \\

Answer: Thank you for this very useful suggestion. I agree that providing a clear step-by-step description of the model would help readers better understand the results.
Following your suggestion, I will add in Appendix a new subsection titled “Algorithmic description of the model” that provides a step-by-step explanation of the simulation procedure. I believe this addition will greatly improve the clarity of the manuscript.  \\

The referee writes:5. Is there any difference in varying the model parameter $\theta$? Does this parameter change result in any significant changes in the results?   \\

Answer: Thank you for this important question. As shown in previous works Petrillo et al. 2020, increasing $\theta$ tends to enhance aftershock productivity and temporal clustering, but does not influence the universal behaviour. The only special case is when $\theta=0$ which is discussed in the appendix. I explained better the role of $\theta$ in the revised version of the manuscript.  \\

The referee writes: 6. Although the author references some research that supports the model results, I suggest that the author should sum up the physical equivalent of the disorder parameter $\sigma$ (similar to my question 1). \\

Answer: We thank the referee for this useful suggestion. Following this comment (and the related comment 1), we have now explicitly clarified the physical meaning of the disorder parameter $\sigma$ in the revised version of the manuscript.  \\

The referee writes: 7. What is the contribution of this research to seismic hazard assessment (expand the ideas of L194-197)? How can seismologists quantify or apply this model to a real seismic sequence?  \\

Answer: I thank the referee for this important point. In the revised version of the manuscript (see Conclusions section), I have now expanded the discussion on the potential implications for seismic hazard assessment. Specifically, our results suggest that fault heterogeneity, which is often poorly constrained in empirical studies, plays a key role in promoting or suppressing foreshock activity. This means that regions with more heterogeneous fault properties may exhibit higher probabilities of detectable foreshock sequences, which can potentially be used as diagnostic precursory signals.
In practice, one promising application would be to combine this modeling approach with independent geophysical proxies for fault heterogeneity - such as fault zone imaging, stress field heterogeneity, variations in seismic velocity, or observations of fault roughness from aftershock distributions. Such proxies could be mapped onto the model’s disorder parameter $\sigma$, allowing a first-order estimate of the expected level of foreshock activity in a given region. Furthermore, the model provides a controlled testbed to quantify how changes in fault heterogeneity affect the likelihood of observable foreshock sequences, helping refine probabilistic forecasting models (e.g., beyond classical ETAS-based approaches). We have clarified these points in the revised version.  \\

The referee writes: Small corrections: Please make the reference format consistent in parts 4 and 5. \\

Answer:  Done. \\