Is the volume-frequency distribution of eruptions a power-law? Accounting for volume uncertainty in modeling the size distribution of volcanic eruptions

Ferrara, Salvatore; Selva, Jacopo; Natale, Jacopo; Marzocchi, Warner

doi:10.5194/egusphere-2026-1415

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https://doi.org/10.5194/egusphere-2026-1415

Preprints

08 Apr 2026

| 08 Apr 2026

Is the volume-frequency distribution of eruptions a power-law? Accounting for volume uncertainty in modeling the size distribution of volcanic eruptions

Salvatore Ferrara, Jacopo Selva, Jacopo Natale, and Warner Marzocchi

Abstract. Forecasting the size of a future volcanic eruption in densely populated areas is a key aspect of volcanic hazard and risk assessment. Estimates of the next eruption's size for both long- and short-term forecasts are typically based on the sizes of past events. Using the erupted volume as a proxy for eruptive size, forecasts are often obtained from the sampling of a power-law distribution. Notwithstanding, the distribution of the measured/inferred erupted volume of past eruptions often appears markedly different from a power-law. Here, we consider how the uncertainty on the volume of past eruptions may affect the shape of a hypothetical power-law distribution. The goal is to understand if the distribution of real data is compatible with an Exponentially Modified Gaussian distribution (EMG) that includes both the power-law and the uncertainty on the observed volumes. We apply this method to two large high-risk calderas, Campi Flegrei, Italy, and Taupo, New Zealand, but it can be potentially applied to any volcano. We find that the EMG distribution provides a good statistical fit to both volcanoes' eruptive records, supporting the use of a power-law distribution for forecasting the volume of the next eruption.

Received: 16 Mar 2026 – Discussion started: 08 Apr 2026

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Salvatore Ferrara, Jacopo Selva, Jacopo Natale, and Warner Marzocchi

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RC1:
'Comment on egusphere-2026-1415', Anonymous Referee #1, 07 Jul 2026
This manuscript presents an analysis of the distribution of the size of past eruptions, expressed in terms of magnitude, aimed at exploring whether a power law distribution for the erupted volumes holds for individual volcanoes. Two volcanoes are considered, namely, Campi Flegrei (Italy) and Taupo (New Zealand). The analysis is clear in its objectives and results: a power law for volumes (which translates into an exponential distribution when magnitudes are considered) appears to be consistent with the data when normally distributed volume uncertainties are added.
I tend to agree with the presented results. Previous work involving one of the co-authors (WM) shows that global volcanic eruptions display (above a certain threshold) a power law distribution of the erupted volumes. If that still holds for individual volcanoes is still an open problem. That is mainly due to the fact that commonly only a limited number of eruptions from each individual volcano is known in sufficient details to be included in statistical analyses. The two volcanoes considered here are among the best studied in the world, and among the few where statistical tests can be applied with some robustness. This work therefore adds an important piece of evidence to the overall puzzle: for two of the best studied volcanoes in the world, the eruption volume distribution is still a power law, as for the global distribution.
That said, I have a number of observations that I believe can contribute to improve the quality of the contribution.
First, the objective expressed in the title is not clear enough. The question “Is the volume-frequency distribution of eruptions a power-law?”, expressed in the title, should make clear reference to the distribution from individual volcanoes. That would eliminate false expectations by the reader on the content of the contribution, and more clearly anticipate its contents.

The authors state that (abstract) “forecasts are often obtained from the sampling of a power-law distribution”, but do not provide any example of that, which would be instead useful.

When referring to the work in Papale et al. (2021) (involving one of the co-authors of this contribution: WM), the manuscript states that eruption volumes have been modelled as being distributed according to a power law “by analogy” with other similar distributions such as the one in the Gutemberg-Richter law for earthquakes. Thai is incorrect. That paper analyses global volcanic eruption databases and finds that global volcanic eruptions are (above a threshold) distributed as a power law – it does not just model them as a power law “by analogy” with something else.

Immediately after (line 46), the manuscript says “Yet it is easy to show that even simple plots of observed eruption volumes sometimes appear to deviate significantly from a power-law”. The “simple plots” in that statement to are almost certainly referred to individual volcanoes – while the reference to Papale et al (2021) immediately before this statement refers to the global scale. As from point 1) above, this contribution appears to frequently mix up the analysis over the global scale with the one in this paper on individual volcanoes. However, the individual volcano scale does not necessarily mimic the global scale (or vice-versa). This manuscript shows that also the individual volcano scale seems to display power law distributions for the erupted volumes. But even if it didn’t, that would not necessarily imply that the global scale does not. The two global and local (individual volcano) scales should be more clearly separated throughout the whole text.

Line 78: the last part of the sentence: “which is also a function of their eruptive style” does not seem to add any relevant information while making the sentence less clear. I suggest to remove it.

Lines 95 and followings: the authors assume that the errors on volume estimates are log-normal distributed with mean on the nominal value. However, the examples they report for potential causes of bad volume estimates (line 85 and followings) all suggest possible dominance of underestimations. Would other distributions, e.g., skewed log-normal, be worth of consideration?

Lines 110-112: the discussion on Mmin here is unclear. The authors say the catalogues are incomplete, therefore, they select Mmin as the value above which the distribution appears to be of the exponential type. Then, if their EMG distribution (the exponential plus log-normally distributed error) fits the data well, they can use it to extend below Mmin to forecast down to the desired M. That discussion seems to ignore the fact that there must be an Mmin below which the distribution is not exponential anymore – which corresponds to a Vmin below which the volume distribution is not anymore of the power law type. Just assuming that they can extrapolate down to any desired M (or V) seems therefore unjustified.

Line 116: the sentence starting with “This volcanic field” is far too long and difficult to read.

Tables 1 and 2. The original data used to construct these Tables are deposit volume. However, neither the deposit volumes, nor the assumed average deposit densities which would allow to retrieve deposit volumes from the reported data, appear in the manuscript. That makes retrieving the original data impossible (the many citations in the table captions do not help: this manuscript should be self-sufficient and present the real data employed rather than their further elaborations, or at least allow the reader to retrieve the real data).

The two figures 1 and 3 are poorly explained, while more clarification seems necessary. Both figures show, for CF and Taupo respectively, the log-likelihood contour plots for the quantities b and Mmin. However, b depends on only Mmin and Mave (line 65). For a given dataset, e.g., the one from either Table 1 or Table 2, once Mmin is defined, Mave is also defined. Thus, it seems that there should be a univocal relationship between b and Mmin for each one of the two CF and Taupo cases, which is not the case in the plots of Fig. 1 and 3. I may misunderstand something here, but in all cases, more explanation would help.

Line 150 and following. The authors may want to clarify, here as well as when describing similar results for the Taupo case, that their first tests with the Lilliefors and AD methods do not involve convolution with a normal error distribution. That should be stated before saying that the corresponding null hypothesis is rejected (which at this stage seems to contradict the results anticipated in the abstract). I could understand that only lines below, after I had already re-read a few paragraphs trying to understand what I was missing in the arguments by the authors.

Lines 157-158. This point is connected to point 9 above. Here the authors say that to calculate magnitudes they start from the DRE and adopt some constant density to convert from mass to volume. However, again, DRE is not the primary data, rather, it is an estimate which is obtained by the deposit volumes – the primary data – after combining it with some assumptions on the average density of the deposit and average density of the pre-eruptive magma excluding gas bubbles. When the authors assume the DRE as the primary data and apply on it some average density to convert into mass (thus magnitude), they are not necessarily consistent with the original deposit volume data - it depends on what assumptions were originally made to estimate the DRE, something which is not said in this manuscript. My strong suggestion is that the authors refer, in the text as well as in Tables 1 and 2, to the original data represented by the deposit volumes, and state what densities they use to convert from the deposit volumes to the erupted masses (the DRE is even not necessary in this analysis).

Line 189: the Ouranui eruption from Taupo volcano is said to have produced “several hundred km3 of products” (citations provided). However, the LaMEVE database reports 1170 km3 of deposits (therefore it classifies that eruption as VEI 8). Could the authors explain such a discrepancy?

Line 208. The authors assume a density for the DRE at Taupo of 2.45 kg/m3; while that same density is assumed to be 2.4 kg/m3 for Campi Flegrei (line 158). In light of the expected significant variabilities concerning the many eruptions from the same volcano, such differences with no explanations provided sound like ad hoc choices decreasing the robustness of the arguments and of the findings. I recommend that more explanation is provided here.

Lines 237-238: the authors say: “In forecasting mode, 𝛽 can be estimated via MLE, while 𝑀𝑚𝑖𝑛 should be defined according to the application's needs.” However, there must be a physical limit in Vmin (thus in Mmin in the exponential distribution), as no power law distribution can continue to indefinitely small values; e.g., for the global distribution of volcanic eruption volumes, Vmin has been found to be close to 170 Mm3 (mean value) (Papale et al., 2021, cited in the manuscript). Stating that Mmin can be defined according to own needs seems therefore unjustified.

Lines 261-262: we are here in the Conclusions section, and the authors write that “the null hypothesis of the magnitude exponential distribution cannot be rejected by the observations”. I suggest to remind here that the magnitude exponential distribution corresponds to a power law volume distribution. That would be useful in general, and guarantee that the readers just interested in the overall conclusions correctly understand the results of this analysis.

Lines 264-266: there is previous literature 1) largely showing that also for volcanic eruptions as for other natural phenomena, low intensity events are more frequent, and 2) referring to continuous distributions to overcome the problems of deriving from discrete eruption classes. As they are presented at the indicated lines within the Conclusions section, these seem to be new findings or novel approaches from this contribution, but they are not. I recommend that this is more fairly stated, and proper references are included.

Lines 307-308 (Appendix): “For magnitudes higher than 5, the simulated samples are left continuous, as the catalogs do not present such discretization above this value”. It would be more correct to say that for such magnitudes the discretization takes place on a finer mesh. There is still discretization, as the deposit volume estimates are unfortunately still rough.
Citation: https://doi.org/10.5194/egusphere-2026-1415-RC1
RC2:
'Comment on egusphere-2026-1415', Anonymous Referee #2, 22 Sep 2026
This article addresses a very important issue in applied volcanology, which concerns the appropriate volume-frequency distribution to use when estimating the likelihood of occurrence of future events of a given size at a particular volcanic system. This is crucial for decision-making with respect to where to focus attention and available resources for risk mitigation and management. It is therefore well suited to the journal.
The study investigates whether uncertainty in eruption volume can be responsible for the apparent deviation of the probability density function from a simple power law. The authors first develop a shifted exponential model for the probability density function (PDF) of eruption magnitudes, based on a logarithmic relationship between magnitude and erupted mass (and hence volume, under an assumed density). They then account for uncertainty in volume estimates by propagating an assumed relative volume error into magnitude uncertainty and convolving the exponential magnitude distribution with the resulting Gaussian error distribution. The authors apply their approach to two well-studied caldera systems, at which a high-magnitude eruption would be highly impactful – Campi Flegrei and Taupo. The study also tries to address the issue of catalogue incompleteness as far as possible, by applying the methodology to different parts of the catalogue (i.e. recent periods of activity with greater likelihood of completeness versus longer time periods with higher likelihood of incompleteness) at each of the case study volcanoes.
The main finding is that, when considering recent catalogue intervals that are less likely to be affected by incompleteness, and when volume measurement uncertainties are sufficiently accounted for, the documented eruption magnitudes at Campi Flegrei and Taupo are statistically consistent with the presented model for the time frames considered. Hence, the presented power-law volume and exponential magnitude distributions could be used to estimate the probabilities associated with different sizes of future eruptions in these systems, respectively. This is an important finding with implications that deserve to be investigated at these and other volcanic systems, including systems that are not caldera-forming. An intriguing aspect of the work is the suggestion of a common or very similar beta parameter for these two systems.
The approach is clearly described and represents a sound framework to approach the problem, in my opinion. The study is well referenced and the figures are of high quality and help to illustrate the conclusions that the authors make, which are appropriate. When assumptions are made, these are clearly stated. I agree with the first reviewer that this is an important study that fills a notable gap, and I would like to see it applied by other volcanologists in other caldera settings as well as other types of volcanic systems to further the discussion on the appropriateness of such forms for the PDFs, and to study the implications.
Below, I add some thoughts to those of the first reviewer:
I agree with the first reviewer that the title should be revisited to clarify the focus of the study on individual volcanic systems; I believe this would also increase readership.

Line 44: I think that ‘By analogy’ could be replaced by ‘Similarly’ and this would not have the unintended meaning noted by reviewer 1.

Line 45: ‘Yet it is easy to show that even simple plots of observed eruption volumes sometimes appear to deviate significantly from a power-law’ – could this statement be supported by an example plot and/or references?

I agree with reviewer 1 that attention should be paid to making explicit when the global or local scale is being discussed. Though I did not find this aspect confusing, it would be good to check this distinction throughout for the avoidance of ambiguity.

Lines 95 onwards: I agree with the question raised by reviewer 1 regarding the assumed form of the error distribution. Would consideration of alternative error distributions substantially affect the conclusions? This seems particularly relevant given the potentially complex and asymmetric uncertainties associated with reconstruction of erupted volumes.

Line 150 onwards: I agree with reviewer 1 that this could be clarified though I did not find this to be confusing.

Line 208: I agree with reviewer 1 on their point about the density assumptions for the two case studies. There will of course be complexity in terms of erupted magma compositions with different densities, and their relative proportions at a given volcanic system, which will require some simplification for this application – however some explanation and justification as to the choices made is required, in my view.

Line 225: ‘In this case, a detailed assessment of catalog completeness is not feasible’. Could the authors specify some reasons for this?
Citation: https://doi.org/10.5194/egusphere-2026-1415-RC2
RC3: 'Comment on egusphere-2026-1415', Álvaro González, 16 Oct 2026

This manuscript presents an interesting and valuable contribution to the size distribution of the volume (or mass) of solid materials erupted by individual volcanic systems, with application to volcanic hazard assessment. It highlights the importance of acknowledging uncertainties in the data when modelling such distributions: if the uncertainties are disregarded, the distribution may seem not to be a power-law, while, if the uncertainties are considered, the power-law size distribution cannot be statistically rejected.
The manuscript can be made suitable for publication with moderate changes. A list of suggestions follows, in order of importance.
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Strengthening the hypothesis that the distribution is a power law
Not being able to reject a power law size distribution of erupted materials is not the same as showing that the power law is the best-fitting distribution. This is not explicitly mentioned in the manuscript.
As discussed in the manuscript, the databases (catalogues) of eruptions for single volcanic systems are still relatively short and incomplete and have large uncertainties. So, simply, the data lacks the statistical power to reject a power law distribution, even if this provides a worse fit than other, competing, distributions with the same number of parameters (something which is not tested in the manuscript).
The global distribution for eruption sizes (merging databases from different volcanic systems) indeed seems to be a power law, as cited in the text. But it is a convolution of (1) the size distribution of the eruptions in each volcanic system and (2) the size distribution of the volcanic systems themselves. (By the way, this is another possible a reason for the different beta values in individual volcanoes and worldwide, to consider in the discussion shown in lines 272-274).
However, to my knowledge, there are other relevant examples of power-law size distributions of eruptions in specific volcanoes:
- In Volcán de Colima the energies of volcanic explosions have been shown to be a power law (exponential distribution of the logarithm of the energies), by Monterrubio-Velasco et al. (2023, 2024).
- In several Japanese volcanoes, the amplitude of ground shaking produced by volcanic explosions also follow power laws (Nishimura et al., 2016).
These explosion databases contain up to thousands of events, so are even orders of magnitude larger than the ones analysed in the manuscript. Albeit they refer to eruption energy, not erupted volume (or mass), it could be argued that emitted energy and erupted volume are expected to be directly proportional to one another. Note that this is not a comprehensive literature list and that other similar publications may exist. But at least citing these robust examples would add support to the choice of a power-law distribution for modelling the size distribution of erupted volumes in an individual volcanic system.
Monterrubio-Velasco, M., Lana, X., and Arámbula-Mendoza, R. (2023). Uncertainties, complexities and possible forecasting of Volcán de Colima energy emissions (Mexico, years 2013–2015) based on a fractal reconstruction theorem. Nonlinear Processes in Geophysics, 30(4), 571-583. https://doi.org/10.5194/npg-30-571-2023
Monterrubio-Velasco, M., Lana, X., and Arámbula-Mendoza, R. (2024). Multifractal structure and Gutenberg–Richter parameter associated with volcanic emissions of high energy in Colima, Mexico (years 2013–2015). Nonlinear Processes in Geophysics, 31(3), 449-461. https://doi.org/10.5194/npg-31-449-2024
Nishimura, T., Iguchi, M., Hendrasto, M., Aoyama, H.; Yamada, T.; Ripepe, M., and Genco, R. (2016). Magnitude–frequency distribution of volcanic explosion earthquakes. Earth Planets Space 68, 125. https://doi.org/10.1186/s40623-016-0505-2
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Distribution biases due to magnitude uncertainties
The manuscript would benefit from better discussing the systematic bias in the exponential distribution of eruption magnitudes due to eruption magnitude uncertainties, using the analogy with earthquakes and the relevant seismological literature.
For example:
Lines 91-92: "This may induce a large number of events to be artificially moved below the threshold defined by 𝑀𝑚𝑖𝑛 , generating significant deviations from the power-law." First, if you are using magnitudes (not volumes), the distribution here would be exponential, not power law. Second, the uncertainty could also move events which are smaller than M_min to an estimated magnitude above M_min. Third, this may shift the exponential distribution but not its exponent by much.
For earthquakes, the systematic bias caused by magnitude uncertainties on the exponential distribution was studied, e.g. by Tinti and Mulargia (1985). They found that, if the magnitude uncertainties are Gaussian (normally distributed), the apparent number of earthquakes exceeding a given magnitude threshold tends to be larger than the real number (the opposite to what you were mentioning for eruptions in lines 91-92).
This bias implies that a correction is needed for the effective number of event counts in the catalogue. In my experience, this correction is widely used in the (stringent) Probabilistic Seismic Hazard Assessments (PSHAs) for nuclear power plants, after its inclusion in the NUREG-2115 report (2012, see chapter 3, page 3-16 and following ones).
So, it would be necessary to better discuss the bias in the distribution resulting from the event magnitude uncertainties, and to compare your approach and this earlier one. I know that Tinti and Mulargia (1985) is, in turn, cited in the work you cite by Marzocchi et al (2020), but I do not see there a discussion on the bias of the event count (as that paper was focused on the exponent of the distribution, which is not significantly affected by this bias unless the magnitude uncertainties are very large).
Also:
Line 95: The assumption of the magnitude errors that you mention (Marzocchi et al., 2020) is the same as Tinti and Mulargia (1985), who would merit to be cited too, I would say.
Tinti, S. & Mulargia, F. (1985) Effects of magnitude uncertainties on estimating the parameters in the Gutenberg-Richter frequency-magnitude law. Bulletin of the Seismological Society of America ; 75 (6): 1681–1697. doi: https://doi.org/10.1785/BSSA0750061681
Electric Power Research Institute (EPRI), U.S. Department of Energy (DOE), and U.S. Nuclear Regulatory Commission (NRC) (2012). Central and Eastern United States Seismic Source Characterization for Nuclear Facilities. NUREG-2115; EPRI Report 1021097; DOE/NE-0140, 6 volumes. EPRI, Palo Alto, CA; DOE and NRC, Washington, DC. https://doi.org/10.2172/1041187.
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Relative error may not be proportional to eruption volume
Line 103: The assumption is made that the volume is affected by a constant relative error e_r = sigma_V / V. This implies that the volume of larger eruptions is measured with larger uncertainty.
But I would say that it may be the opposite, right? Deposits of small eruptions are more difficult to measure and can be eroded or covered more easily than the deposits of larger eruptions. Indeed, it is what you later argue, e.g. for the earlier part of the record at Campi Flegrei. So, the volumes of large eruptions are measured, I would say, with lower relative error. Please reconsider if your assumption is justified and modify the analysis accordingly if needed.
An alternative might be to consider that sigma_V is constant (so that a smaller V would have larger relative error), but with sigma_V limited so that the error is not larger than V itself.
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Justification of the use of eruption magnitude instead of volume
30-39: Erupted volume (or mass) is a meaningful physical magnitude, while eruption magnitude is a fashionable arbitrary definition. Please better justify your choice of using eruption magnitudes in the modelling instead of volume values directly (even if they span several orders of magnitude).
I think that eruption magnitude becomes useful for you approach in the sense that it will be exponentially distributed (allowing a classical Lilliefors test for the exponential distribution). And also, because magnitude (as for earthquakes) is a handy, “compact” number. But using such a logarithmic scale may cause trouble e.g. for hazard communication (as for earthquakes, where non-technical audiences struggle to understand, e.g. that a magnitude 7 earthquake is indeed ~32 times larger than a magnitude 6 earthquake).
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Wording suggestions and minor issues line by line
9: "in a densely populated area" (singular, given that you are referring to an eruption).
32: "log" (lowercase), or better "log_10", if you wish, to make it crystal-clear.
33-34: "events that span several orders of magnitude in volume"
41: "captured by" -> "described by", "expressed as", etc.
43: roll-offs (plural), since you are mentioning two.
56: I would place the reference (Kagan, 1993) at the end of the next sentence, after "seismic moment", since it would rather belong there.
59: "shifted" -> This wording referring to the distribution may not be accurate in this context, I would say, albeit it can make sense when discussing the exponentially modified Gaussian distribution. Consider replacing "shifted" by:
- "left-truncated (or lower-truncated)": Observations below a specified threshold are excluded from the dataset. For example, eruption volumes below a minimum detectable volume are not included.
Or:
- "left-censored": Observations below a threshold are known to exist, but their exact values are unknown or recorded only as being below that threshold.
61-62: catalog -> I would say "eruption catalog", instead of just "catalog" at least once. Also, beware if you are using English spelling (catalogue) or US spelling (catalog) to match the rest of the text.
101-102: Please mention that erfc is the complementary error function.
124: The acronym "DRE" is not defined in the manuscipt.
138-139: log-likelihood (lowercase)
146: Pistolesi et al. (2017) is cited here but not in the Table 1 caption, which is confusing. Which data were updated following them?
180-181: Figure 2b: The empirical cumulative distribution function should reach 1.0 if properly normalized, but here it does not reach it. Also, please show the plot only from Y=0 to Y=1 as it is the only possible range. Better I would use the classical term "empirical cumulative distribution function" instead of "empirical cumulative density function", which is generally considered incorrect terminology.
206: better “Taupo eruption magnitudes”
221: "The null hypothesis of EMG distribution is rejected only when e_r = 0.25." Better: "Among these cases, the null hypothesis of EMG distribution is rejected only when e_r = 0.25."
224-225: log-likelihood (lowercase)
230-231: Figure 4b: The empirical cumulative distribution function should reach 1.0 if properly normalized, but here it does not reach it. Also, please show the plot only from Y=0 to Y=1 as it is the only possible range. Better use the classical term "empirical cumulative distribution function" instead of "empirical cumulative density function".
235: "volume or magnitude". Better: "volume or magnitude, respectively"
239: "or the minimum magnitude capable of reactivating the volcanic system". I would think that, if there is an eruption of whatever M in a volcano, the system has been activated (by definition). What do you mean here?
282: "empirical cumulative density function" -> I would say "empirical cumulative distribution function"
314: “Model code available upon request to the authors”. Placing it in a public repository would benefit your readership and would make your work more reproducible and citable.
339-340: In the original title, "goodness of fit" was written within quotes (Asymptotic theory of certain "goodness of fit" criteria based on stochastic processes).
340: DOI is missing: https://doi.org/10.1214/aoms/1177729437
341: DOI is missing: https://doi.org/10.1080/01621459.1954.10501232
352-359: The references starting by Bonnadona are not in proper alphabetic order.
394: DOI is missing: https://doi.org/10.1080/01621459.1967.10482916
396: DOI is missing: https://doi.org/10.1080/01621459.1969.10500983
406-419: The references starting by Natale are not in proper alphabetic order.

Citation: https://doi.org/10.5194/egusphere-2026-1415-RC3

Salvatore Ferrara, Jacopo Selva, Jacopo Natale, and Warner Marzocchi

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Short summary

We develop a probabilistic framework for modeling volcanic eruption sizes. Assuming a power-law distribution on erupted volumes, we analyze how measurement error can cause the observed trend to deviate from such a distribution. We apply this framework to Campi Flegrei, Italy, and Taupo, New Zealand, and observe that when the error is properly accounted for, the data distribution is compatible with a power-law, supporting the use of such a distribution in size forecasting.


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