the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 19 Jun 2026
Earthquake Scaling Equations Under Small Strain, Steady Moment Release-Rate Conditions in Southern Andes from 2015 to 2017
Abstract. In the South Andes western edge, a very active seismic contact, with earthquakes up to magnitude 9.5 and ca. 4000 km in length threatens cities and very large populations. The existence of modern seismological networks along the contact allowed the observation of unprecedented earthquake cycle characteristics, which can improve our ability to estimate earthquake hazard, a main objective of seismology. Using dimensional and similarity analysis techniques, we show precise mechanical conditions under which the earthquake generation process unfolds, and theoretically-derive a set of scaling equations linking renormalized variables. Later on, we test our theoretical results using a curated earthquake point-catalog by using gridding, box-counting, statistical bootstrap and fixed-point iteration collapse techniques. We found non-trivial scaling laws valid across multiple orders of magnitude capable of describing a complex interplay between renormalized earthquake occurrence and renormalized seismic-moment release-rate. We discuss implications in terms of small-strain and seismic-moment release-rate imposed; cutoff magnitudes, statistical properties of seismicity, how seismic cycle might be analyzed in presence of long-term correlations, seismic-moment transfer under small-strain conditions, earthquake hazard implications and tectonic status. Finally, we conclude that exponents characterizing seismicity are related through a set of scaling equations, meaning that all considered processes have very long-term correlations. The available data suggests a single power law fitting data across the western edge. These equations were obtained by an asymptotic analysis, also a cascade mechanism is proposed to explain the observed moment release behavior.
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Status: open (until 23 Sep 2026)
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RC1: 'Comment on egusphere-2026-3399', Anonymous Referee #1, 20 Jul 2026
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AC1: 'Reply on RC1', Patricio Toledo, 31 Aug 2026
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On behalf of the co-authors, I would like to thank the reviewer for his comments. To encourage discussion, we will respond to the comments point by point, indicating the changes and clarifications we will incorporate into the document. A formal response letter and updated document will be sent for revision also.
Major issues
1) We use the term “seismogenic thickness” in a rather informal way, without intending to refer to the length defined previously by Sibson, R. (1982, 1983, 1984). We would like to note that at the end of the paper, in the discussion section, we propose a scale that is approximately equal to 30 km, in line with the length indicated by the reviewer. When performing the dimensional analysis, prior to any calculations, we use a large distance, approximately equal to the catalog spatial width; the same applies to the observation time. We will clearly indicate the meaning of H and include the observation by Sibson, R. (1982, 1983, 1984) in the discussion section.
2) Thank you for the observation. We will add an appendix with the complete algebraic derivation leading from (10) to (12) via (11).
3) We appreciate the comment. Northern Chile is one of the most seismically active regions on the planet. We believe that the general characteristics of the energy dissipation process, as observed through the statistical properties of the point catalog, are sufficiently robust. In that regard, we know that the catalog contains information on the aftershocks of the 2015 Iquique earthquake and likely other earlier events; however, the cell-based analysis method, in conjunction with the similarity conditions, naturally achieves “declustering.” We will include these observations in the discussions and expand the text.
4) Thank you for the comment. We will replace “small, medium, large” with the precise numerical intervals.
5) We appreciate the comment. The inter-event times and the occurrence times of the aftershocks in Omori’s law are related; we refer to this in Section 2.3. We have not been able to derive Omori’s law from first principles, which
is reflected in the separation between the system of equations (12), where there is no geometric relationship between inter-event times and the spatial set defined by the hypocenters, as one would suppose given that aftershocks are confined to rupture area. Therefore, we will include the hypothesis proposed by the reviewer in the discussion, since it indeed opens up interesting perspectives. Although, we would like to clarify that interevent times and Omori times are not the same.6) We appreciate the comment. The Gutenberg–Richter law is paired with the fractal distribution of hypocenters as observed by Aki, K. (1981) and in our text in Eq (12). Therefore when cell proper-length is included in moment statistics, the variability is smoothed out, as can be seen in Figure 7. We will explicitly add this observation to the text.
7) We appreciate the observation. The value of Q is nominal, obtained from the literature. Figure 7 suggests, however, that it is a reasonable value that allows us to obtain a power-law for the renormalized variables. We will indicate in the discussion that the estimation of this parameter is an area for future study.
8) Thank you for the comment. We will try to include a better explanation in the paragraph describing the fixed-point iteration. According to studies in statistical physics (McComb, W. 2003), the equilibrium surface of nonlinear systems with many components has a multitude of critical points, which manifests itself, for example, in the phenomenon of frustration—that is, the inability of these systems to reach a global minimum, remaining forever trapped in local conditions, in quasi-static equilibria.
9) Thank you for the observation. We will include a paragraph using the proposed analysis. It can be seen in Figures 5 and 7 that the greatest effects are due to the long inter-event times and the long distances covered by the catalog.
Minor issues:
1) We will try to improve the colors in the figures
2) Thank you for the comment. We will change the lines from solid to punctured.
Issues with English writing
1) Thank you for the comment. We will improve the abstract.
2) Thank you for the comment. We will revise the long paragraphs to ease reading.
3) Thank you for the comment. We will consult with the editors on the exact way to standardize the document.
4) Thank you for the comment. We will apply these changes.
5) Thank you for the comment. We will apply these changes.
6) Thank you for the comment. We will apply these changes.
7) Thank you for the comment. We will review it.
8) Thank you for the comment. We will update the sentences as suggested.
Citation: https://doi.org/10.5194/egusphere-2026-3399-AC1
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AC1: 'Reply on RC1', Patricio Toledo, 31 Aug 2026
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My comments on the manuscript, entitled ‘ Earthquake Scaling Equations Under Small Strain, Steady Moment Release-Rate Conditions in Southern Andes from 2015 to 2017’ (egusphere-2026-3399 (NPG)) authored by Patricio A. Toledo, Cristián Siegel, Benoit Derode, Raúl Madariaga, and Jaime Campos
Although the study is interesting and significant, I cannot correctly evaluate whether the manuscript may be accepted for publication in its current form or not due to the following reasons shown in ‘Major Problems.’ These reasons make me unable to judge whether the statistical results made by the authors are physically significant enough. Hence, I suggest that the authors must answer the questions. This manuscript should be substantially revised and then reviewed again.
Major Problems:
(1)In Figure 1 (or on Line 136), the authors seem to define the seismogenic thickness (denoted by H) from the ground surface to the depth where the deepest earthquake happened on the subduction zone. This definition is clearly different from the common one proposed by Sibson (1982, 1983, 1984). From Sibson’s model, the earthquakes happen at the upper layer of the crust with a temperature of lower than 400±50oC in the continental plate and at the uppermost layer of the subduction zone. Considering the commonly-used seismogenic-zone depth, H should be shorter than 33 km. Hence the plots for H>33 km in Figure 6 are insignificant. On the other hand, in case that H represents the deepest depth of earthquake occurrences, the results in Figure 6 are OK. The authors should clearly explain the definition of H.
References
Sibson, R.H. (1982). Fault zone models, heat flow, and the depth distribution of earthquakes in the continental crust of the United States. Bull. seism. Soc. Am. 72, 151-63.
Sibson, R.H. (1983). Continental fault structure and the shallow earthquake source, J. Geol. Soc. London, 140, 747-767.
Sibson, R.H. (1984). Roughness at the base of the seismogenic zone: contributing factors. Journal of Geophysical Research, 89, 5791-5799.
(2)The author should clearly explain the way how to obtain the results shown in Eq. (12) from Eqs. (9) and (11)?
(3)In the study area, a large number of larger-sized (M>7) earthquakes occurred before 2015 as described from Line 232 to Line 238. In this study, the authors only used the earthquakes that occurred from 2015 to 2017. Can the results inferred from such a data set represent the statistical characteristics of larger-sized earthquakes in this area?
(4)Could the authors explain the magnitude ranges of small, medium, and large earthquakes in Figures 5–9?
(5)The distribution of data points of Π versus ΠT as displayed in Figure 5 seems able to be described by the Omori-Utsu’s law of aftershocks (e.g., Omori, 1894; Utsu, 1961; Utsu et al., 1995). In the subsection of ‘3.1 Southern Andes tectonic framework,’ the authors reported the occurrences of the great 2015 Mw8.3 Illapel earthquake the 2016 Mw7.6 Chiloé earthquake in the study area. Could a large number of events used in their study be the aftershocks of the two large earthquakes? The authors should discuss the problem deeply.
References
Omori F. (1894). On the aftershocks of earthquakes. J. Coll. Sci. Imp. Univ. Tokyo, Japan, 7, 111-200.
Utsu T. (1961). A statistical study on the occurrence of aftershocks. Geophys. Mag., 30, 521-605.
Utsu T., Y. Ogata, and B.S. Matsu’ura (1995). The centenary of the Omori formula for a decay law of aftershock activity. J. Phys. Earth, 43(1), 1-33.
(6) Gutenberg and Richter (1944) proposed the frequency-magnitude (FM) law: log(N)=a-bM where M is the earthquake magnitude, for example, the surface-wave magnitude (Ms), N is the cumulative number of events with magnitudes≥M and a and b are, respectively, a constant and the scaling exponent. The relationship between the seismic moment, Mo, and the surface-wave magnitude, Ms, is: log(Mo)=1.5 Ms+16.1 (Purcaru and Berckhemer, 1978). From the two equations, we may obtain the following expression: N~Mo-2/3. In Figure 6, the two normalized parameters Π and ΠMo are related to N and Mo, respectively. But, the distribution of data points in Figure 6 could not described by this expression. Could the authors explain the discrepancy?
References
Gutenberg B. and C.F. Richter (1944). Frequency of earthquakes in California. Bull. Seism. Soc. Am., 34(1), 185-188.
Purcaru G. and H. Berckhemer (1978). A magnitude scale for very large earthquakes. Tectonophysics, 49, 189-198.
(7)In Figures 8 and 9, the authors plotted the data points of Π and ΠQ. The authors defined the parameter Q to be the seismic moment release rate. However, they did not clearly explain how to estimate the value of Q. Is the value of Q=1.5×1012 W acceptable for the study area?
(8)Figures 8 and 9 show negative correlation between the two dimensionless quantities, i.e.,Π and ΠQ. In Figure 8, as gQ≤0 the data points are clearly separated into several groups and each group shows a negative correction between Π and ΠQ. On the other hand, as gQ>0 the data points are clustered together to form a single group and the degree of dispersion of data points decreases with increasing gQ. The authors should explain the internal mechanism.
(9)From the plot of Figure 9, the authors also inferred the correlation in the form:Π* ~ (ΠQ*)-0.835. The normalized parameterΠQ is defined to be Q/(Dsl3t-1). Since Q is taken to be a constant of 1.5×1012 W in their calculations, Π* is correlated inversely with Ds and l3 and correlated with t. Hence, the power law with a negative exponent of -0.835 between Π* and ΠQ* could be produced by either Ds or l. The authors should explain which parameter is the major one in yielding the power-law correlation.
Minor Problems:
Problems with Figures:
(1)The quality of figures is not good enough for readers. For some figures, the colours of data points are not consistent with those in the legend.
(2)Since the b-value is always positive from observations, it would be better to plot the dashed lines or dotted ones for the b–d relationships as b<0 in Figure 2.
Problems with English Writing:
(1)The abstract is not concise.
(2)The English writing is good. Nevertheless, there are still some typo errors. In addition, some sentences are too long to easily read.
(3)The authors used ‘equation (i),’ ‘Equation (i),’ ‘Eq(i)’, ‘ and ‘the Eq. (i)’ to represent an equation. They should unify the notation.
(4)On Line 104: ‘The statement ‘(about 0.1 and 1.0)’ should be ‘(0.1 and 1.0, respectively).’
(5)From the definitions of ‘l and ‘m’ in the first paragraph of the subsection ‘2.1 Complete similarity condition, it is better to re-write Eq. (3) as ‘Π=Φ(ΠT,ΠMo,ΠH, ΠQ,Πu),’ because ‘Πu’ is not taken into account in this study.
(6)On Line 127: The statement ‘Golitsyn (2007, 2001)’ should be ‘Golitsyn (2001, 2007).’
(7)In Table 2, the unit of interevent time, t, is ‘seconds.’ Why the authors type ‘ms’ in the table?
(8)On Line 302: The authors wrote the statement: ‘Figure 5 shows cell maximum dimensionless interevent frequency ΠT versus dimensionless event number Π.‘ Although this statement is OK, I assume to re-write it as ‘Figure 5 shows dimensionless event number Π versus cell maximum dimensionless interevent frequency ΠT.’ This is due to a fact that Π and ΠT are along the vertical axis and horizontal one, respectively. The similar statement also appears for Figures 6–9.