the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Conversion relationships between Modified Mercalli Intensity and Peak Ground Acceleration for historical shallow crustal earthquakes in Mexico
Abstract. New empirical relationships between modified Mercalli intensity (MMI) and synthetic peak ground acceleration (PGA) are developed for shallow crustal earthquakes in Mexico. Ground motion data from 18 moderatetolarge earthquakes (4.5 < M_{W} < 7.5) and the corresponding 531 MMI information reports were employed. Synthetic PGA data were generated using the finitefault stochastic method considering different rupture scenarios in order to extend the limitations of the dataset. Linear and bilinear regression techniques were used considering a binning averaging procedure and the whole dataset. On one hand, a set of MMI predictive equations independent of moment magnitude (M_{W}) and hypocentral distance (R) were derived. Despite weak dependencies of the residuals on M_{W} and R terms, on the other hand, we also developed refined predictive relationships that include these parameters as independent variables. The refined PGA to MMI conversion equations show slightly less variability than simple linear equations in predicting intensity values. The proposed predictive equations are consistent with similar relationships in other regions of the world. The discrepancies among the different relationships may reflect the differences in input data, particularly related to the macroseismic intensity assignments, which are inherently subjective, and the tectonic regime. The proposed relationships can be used for improved hazard assessments in Mexico.
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Status: open (until 24 Jul 2024)

RC1: 'Comment on nhess202492', Anonymous Referee #1, 20 Jun 2024
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Comments to authors
This manuscript is well written and easy to read. The new empirical relations proposed will be useful for future studies. The only weakness that I find in this study is the lack of actual PGA observations that could be useful to calibrate the stochastic models. However, the parameters used with EXSIM12 are adequate and provide conservative estimates of PGA.
A few suggestions follow:
Pg.3, line 24: the 1887 event is better known as the Sonora earthquake.
Pg.6, line 18: I think the year of the reference is 2015.
Pg.7, lines 1011: Are the stress drop values (1, 5, 10 and 20MPa) typical or expected for the earthquakes modeled?
Pg.7, line 17: is there any observed PGA information from any of the earthquakes analyzed or from other similar events?
Pg.9, lines 1920: “The bilinear models differ significantly at lower PGA values” Why is that?
Comments to Editor
This is an interesting and well written paper that provides new empirical relations between peak ground acceleration and modified Mercalli intensities which may be useful for hazard assessments in central and northern Mexico.
I recommend the manuscript is accepted for publication in Natural Hazards and Earth System Sciences.
Citation: https://doi.org/10.5194/nhess202492RC1 
RC2: 'Comment on nhess202492', Anonymous Referee #2, 01 Jul 2024
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RodríguezPérez and Zúñiga have proposed PGA to MMI relationships for Mexico, by using observed MMI from historical events, and synthetic PGA simulated using stochastic finitefault method. The author’s PGA simulation approach appears comprehensive. This study after some improvements, could be a good contribution to the seismic research community.
However, I have some major concerns in the study in its current form, especially relating to the statistical analysis. One of the primary challenges for this study is not being able to fully validate its results due to lack of ground truth PGA observations. I have described all my concerns below. As line numbers restart on each page, I have used the notation pagenum.linenum.
 General  The problem that the authors are trying to solve has not been defined anywhere, including the abstract and introduction. They mention that the goal of the study is to develop new empirical relationships between MMI and PGA in Mexico. But why is it necessary to develop these relationships? It would be helpful to include text to motivate the problem that the authors are trying to solve.
 General  One sentence that could be the potential motivation for this study is Line 10.13 where the authors mention that reported MMI could be used as a proxy for estimating PGA for historical earthquakes using their developed relationships, in order to better understand seismic mechanisms in Mexico, and eventually estimate better hazard. However, throughout the regression analysis, the authors have used MMI as the dependent variable and PGA as the independent variable, which would not meet this objective of developing a relationship to convert MMI to PGA. Instead, the regression analysis should be done with PGA as the dependent variable.
 Line 1.21  I could not find support in the text for this statement in the abstract  The refined PGA to MMI conversion equations show slightly less variability than simple linear equations in predicting intensity values. Additionally, it is not clear what the authors are referring to as “refined” and “simple” equations in the statement.
 Line 1.25  The statement could be made better by adding how “the proposed relationships can be used for improved hazard assessments in Mexico”.
 Line 3.3  Suggest changing sentence for clarity  Various studies have proposed relationships between seismic intensity (measured by MMI, DYFI, JMA, etc.) and ground motion parameters (measured by PGA, PGV, Sa, etc.) across distinct geographical regions.
 Line 3.11  Since the last M ≥ 6 event *that* originated in north…
 Fig 1  The dark gray shading, dashed lines, and line with triangles on the map are undefined.
 Fig 1  The figure can be improved by changing the color/style of earthquakes in each of the three regions, in order to easily associate events with their respective regions.
 Fig 1  The figure can be further improved by using a topographic map as base layer.
 Paragraph 4.19  The information currently provided for observed MMI reports is insufficient. Please provide more information about how MMI was calculated for historical earthquakes, especially those dating back to the 1500’s, and the sources of original data for these MMI reports.
 Paragraph 4.19  It would be helpful to include a figure with the count and range of MMI reports obtained for each of the 18 events used in this study. Although this could be roughly interpreted by combining Fig 2 and Table 1, adding a separate figure will both improve the data section, and make it easier for the reader to understand the observation data.
 Line 4.20  Suggest specifying “moment magnitudes”.
 Eq 2  Suggest including the left side of the equation. Y = …
 Eq 2  kappa is undefined.
 Eq 2  G(r) is undefined. Perhaps, it’s supposed to be G(R).
 Line 5.23  R_{theta phi} is not present in the equation. theta and phi are undefined.
 Line 6.4  Suggest defining i, j prior to this line for better clarity.
 Eq 3  Please define the difference between lowercase and uppercase `a`, and the relationship between them, if any. Lowercase a(t) is defined as the sum of sub fault contributions and it is not clear which contribution it refers to.
 Line 6.15  Is the time delay relative to the sub fault i=0, j=0?
 Line 6.15  I am confused about the notation i, j. In eq 3, it is used as sub fault along the length and width of the rupture plane respectively, but has not been defined as such in previous equations. Are these i, j different than the ones in eq 2?
 Line 6.16  I am not familiar with the term “rise time”. Could this be defined?
 Line 7.4  It will be helpful to mark the 1887 event on the map in Fig 1 for better understanding of its location with respect to other events.
 Line 7.11  From Figs 3 and 4, it appears that the PGA for different stress drops was calculated separately and regression equations were developed for each stress drop. It would be helpful to clarify that here, and also add why multiple values were selected, compared to all other variables, for which the best single value was chosen.
 Line 7.22  It is not clear where two branched trend was “previously reported”.
 Line 7.23  The difference between Figs 3 and 4 is never described.
 Line 8.2  MMI defined up to two decimal points indicates high precision in MMI reporting, which seems unlikely given the historical nature of most events. My previous comment about providing more information about MMI calculation might help here. Overall, it would be helpful to understand the expected variability in the MMI estimates from the reports.
 Paragraph 7.17  The statements regarding the two methods  using complete dataset and MMI bucketaveraged datasets are mixed together in this paragraph, making it harder to understand which model the authors are referring to in a given sentence. Suggest splitting the two models in different paragraphs for clarity.
 Eqs 4 and 5  The number `1` in t_1 indicates presence of t_x. Suggest removing the numeral since there is only one threshold in the two branch model.
 Eq 5  logPGA is not in the equations but is used in the threshold t_1. Instead, since the correction term is a function of MMI, the threshold should be changed to MMI in this equation. For example, delta_MMI (MMI) = … for MMI < x.
 Eq 5  If the MMI is near threshold t_1, is it possible that applying the correction would move the MMI to a different branch? It would be helpful to clarify this and how to resolve this situation.
 Line 8.20  The statement is unclear, and I am not sure how the models were combined.
 Tables 3, 4  Please define standard errors in equations 4 and 5 as it is difficult to associate them to the respective equations in their current form. For example, MMI=c1+c2 logPGA + epsilon, where epsilon ~ N(0, sigma_ij)
 Line 9.3  Please describe how the “evaluation” was done. There appears to be lower residual variability at lower magnitudes, and both low and high hypocentral distances from Fig 5.
 Line 9.5  The sentence is not clear. What is meant by residuals are smaller by factor of 8  is it the variability in residuals, or the extreme residual value from bilinear model compared with that of the averaged model?
 Line 9.5  Since there is only one data point per MMI (due to averaging PGA) for model 2, the comparison between models 1 and 2 cannot be made directly. The variability of residuals given M, R will always be zero or close to zero for model 2 due to the very few data points. In fact, it is more likely that the mean of residuals is nonzero for model 2 given the scarcity of data points, thus nullifying the assumption of residuals following zeromean normal distribution.
 Figs 710  It would be better to combine subfigures a and b to be able to compare the results of this study with existing relationships across all proposed stress drops. In fact, it may be helpful if the subfigures were shown for bilinear and linear models instead, so they can be compared easily. Another option will be to show 1 figure per stress drop with both models in the same figure.
 Line 9.16  Why were these specific models chosen? Please provide information about comparative model selection based on similar magnitudes/distances/rupture mechanisms/site conditions, etc. In addition, please add a statement about which of the comparative relationships is expected to be the closest to that of this study based on rupture and ground motion attenuation characteristics in Mexico, to make it easier for the reader to focus on the differences with that model.
 Line 9.20  It appears that the difference of bilinear models occurs at t_1 which happens to be close to 10 cm/s2. Suggest modifying this statement to indicate that the lower branch of the bilinear model (approx < 10cm/s2) deviates from other existing relations.
 Line 10.1  I disagree with this and the following statements, as the standard error between models 1 and 2 cannot be compared directly due to the large difference in the number of points used for regression in both models.
 Line 10.9  This statement may need to be revised based on my previous comment about observing some trends in residuals over magnitudes and distances from the figures.
 Line 10.13  The approach taken by this study is interesting given the authors’ statement that the objective is to convert reported MMI to PGA/PGV, instead of converting PGA to MMI. In that sense, why have the authors chosen to use MMI as the dependent variable in this study? Note that linear regression does not give the same result by reversing the dependent variable since the residual square error is minimized for the dependent variable only. In that sense, in order to create an appropriate relationship to convert reported MMI to PGA, the authors should revisit their approach and fit regression equations with PGA as the dependent variable.
 Line 10.18  Please include hypocentral distance for the MMI observation.
 Line 10.21  I disagree with the statement that “stochastic ground motion simulation method is a valid methodology”, since it indicates to the reader that this may be one of the most appropriate approaches to solve this problem. In fact it is not possible to evaluate the validity of this simulation approach with other simulation approaches since there is no ground truth data. In that sense, it is completely possible to use an entirely different approach (e.g., a simplified GMPE with point sources) to generate synthetic PGA, and then fit regression equations between those PGA and observed MMI. Without ground truth, there would be no way of knowing which of the two approaches are better suited. No doubt that this study is comprehensive and beneficial for pushing the boundaries of seismic research in Mexico, however, the authors have only proposed one of the possible valid methodologies, the validity of which cannot be determined given the study’s lack of ground truth MMI + PGA observations at the same sites.
Citation: https://doi.org/10.5194/nhess202492RC2 
RC3: 'Comment on nhess202492', ChungHan Chan, 05 Jul 2024
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The manuscript entitled ‘Conversion relationships between Modified Mercalli Intensity and Peak Ground Acceleration for historical shallow crustal earthquakes in Mexico’ proposed new relationships between modified Mercalli intensity and peak ground acceleration (PGA) for shallow crustal earthquakes in Mexico. Such a study could be crucial for subsequent research on seismic hazards and for determining strategies for hazard mitigation. After read through this manuscript, I have a number of questions and comments about the manuscript, listed below.
Source parameter uncertainty: Earthquake simulation requires several parameters. Although the rupture parameters for each earthquake was shown in Tables 1 and 2, their reliability and uncertainty were not discussed. I also noticed that not all of parameters shown in equation (2) were specified in the text (e.g., Swave velocity, corner frequency). Furthermore, it is rather difficult to obtain rupture parameters for historical events (without instrumental records).
Simulation reliability: The uncertainties of the source parameters were not appropriately addressed, and the reliability of the simulations was also not thoroughly discussed. While considering different stress drops is a good approach, the study only presented the results with the lowest deviation for the regression with intensity. In my opinion, this comparison alone cannot accurately identify the proper stress drop. Instead, I suggest obtaining the stress drop for some events from previous studies or waveform inversion. Additionally, there is no discussion on confirming the simulation's reliability. I recommend comparing the simulated PGA with some instrumental observations to validate the simulations.
Path and site effect: Equation (1) represents the source, site, and path components in the waveform simulation. While the source component has been described in the text, the path and site effects have not been specified. Which velocity model was implemented, and are the simulation results sensitive to the velocity model? Although this study proposed simulation sets incorporating both rock and soil conditions, the epistemic uncertainty could be further minimized by considering the specific site conditions for each location.
Citation: https://doi.org/10.5194/nhess202492RC3
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