| 17 Jul 2026
Observing Phase Transitions in Extreme Rainfall Events: Insights from Sub-Minute Observations and Universal Multifractal Analysis
Abstract. Complex systems across scientific disciplines often undergo abrupt, qualitative reorganizations –referred to as phase transitions– in which their fundamental properties change discontinuously. In geophysical contexts, such transitions can appear as sudden alternations in statistical behavior or scaling characteristics. In this study, phase transitions in extreme rainfall events are investigated through the Universal Multifractal (UM) framework, using ultra-high-resolution (10-second) disdrometer measurements collected during three typhoons in 2022: Hinnamnor, Nesat, and Nalgae. Two categories of multifractal phase transitions are examined: (i) transitions associated with sampling limitations and (ii) those related to moment divergence. While sampling limitations were observed during Typhoons Hinnamnor and Nesat, Typhoon Nalgae exhibited a clear case of moment divergence –a rare phenomenon indicative of extremely concentrated rainfall. These findings highlight the essential role of ultra-high-resolution time series in detecting and characterizing extreme hydrometeorological events. The study further demonstrates the capability of the UM framework to capture complex rainfall dynamics and offers insights that may contribute to improved hydrological modelling and flash-flood forecasting.
Competing interests: At least one of the (co-)authors is a member of the editorial board of Nonlinear Processes in Geophysics.
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General
This article examines the statistical properties of extreme precipitation observed during typhoons, using measurements taken by an ultra-high-resolution disdrometer. The authors use the ‘Universal Multifractal’ (UM) model to identify phase transitions associated with the divergence of statistical moments, with the aim of improving our understanding of rainfall variability at different scales, and in particular of extreme rainfall. Understanding the scale properties of extreme precipitation remains a major challenge, particularly in the context of increasingly frequent high-impact rainfall events. The availability of ultra-high-resolution observations provides a valuable opportunity to study small-scale precipitation variability.
However, in its current form, I believe the manuscript would benefit from substantial revision before it can be considered for publication. My main concerns relate to the originality of the work, the lack of references to previous studies, and the robustness of the statistical analysis
Major Comment
Authors should explicitly situate their work within the existing literature and explain what new insights have emerged from the present analysis. Without a clearer positioning, the novelty and scientific contribution remain difficult to assess. An important limitation of the manuscript lies in the lack of discussion of previous studies that have examined the multifractal properties of precipitation using high-resolution disdrometer observations. For example, UM modelisation of disdrometer observations have been discussed in the following studies that are not mentioned in the state of the art :
De Montera, Barthès & Mallet (2009), The effect of rain-no rain intermittency on the estimation of the Universal Multifractal model parameters, Journal of Hydrometeorology.
Seppe Silva, M., Vieira Casanova Monteiro, R., Jose, J., Gires, A., Paz, I., Tchiguirinskaia, I., & Schertzer, D. (2025). Multifractal comparison of rainfall measurement with the help of a disdrometer and a mini vertically pointing Doppler radar. Hydrological Sciences Journal, 70(7), 1072-1090.
Verrier, De Montera, Barthès & Mallet (2010), Multifractal analysis of African monsoon rain fields, taking into account the zero rain rates problem, Journal of Hydrology.
Verrier, Mallet & Barthès (2011), Multiscaling properties of rain in the time domain, taking into account rain support biases, Journal of Geophysical Research.
Zero rainfall values are likely to negatively impact the scaling properties of rainfall, making it difficult to retrieve reliable model parameters in the first place (de Montera et al., 2009; Verrier 2010; Veneziano and Lepore, 2012; Mascaro et al., 2013) and other publications by the authors themselves. The presence of zero rainfall values (rain/no-rain intermittency) and rain support effects introduce significant biases in the estimation of Universal Multifractal parameters if they are not explicitly taken into account. Given that this article deals with the estimation of high-order moments and their potential divergence, this issue must be addressed and its potential impact discussed .
Veneziano, D., & Lepore, C. (2012). The scaling of temporal rainfall. Water Resources Research, 48(8).
Mascaro, G., Deidda, R., & Hellies, M. (2013). On the nature of rainfall intermittency as revealed by different metrics and sampling approaches. Hydrology and Earth System Sciences, 17(1), 355-369.
Moment divergence estimated from finite datasets is influenced by several factors, including finite sample size, rain intermittency, support effects, estimator bias, and measurement uncertainty. In the present study, the statistical representativeness of the analysis is also limited by the relatively small observational sample, consisting of only three locations and slightly more than 100 hours of measurements in total. Such a limited dataset makes it difficult to distinguish a robust physical transition in rainfall organization from an event-specific or sampling effect.This issue is important when considering higher-order moments, which are dominated by intense but rare fluctuations and are therefore highly sensitive to the number of independent samples available. As the authors themselves point out in the conclusion, a more extensive dataset—including more extreme events, different climatic regimes and multiple observation sites—would be required to ensure the robustness of the conclusions regarding these multifractal phase transitions. The authors should therefore present the detected divergence as an indication of a possible multifractal transition rather than as its proven physical manifestation
The conclusions appear to overstate the physical significance of the results. In particular, the statement that"The divergence observed in Nalgae reflects a genuine physical realization of unbounded rainfall intensity rather than a mathematical artifact." is not sufficiently supported by the analyses presented. This sentence seems to suggest that the rain observed during Nalgae may indeed have reached a physical regime in which the intensity can become unbounded, and that the divergence is not an artefact linked to the data or the method. From a physical standpoint, rainfall intensity is ultimately constrained by atmospheric dynamics and microphysical processes. Therefore, the observed divergence should be interpreted as a statistical property of the rainfall distribution rather than as evidence that rainfall intensity itself is physically unbounded It means that, within the range of scales and samples observed, the distribution of intensities exhibits behaviour consistent with a heavy-tailed distribution for which certain theoretical moments are undefined. The divergence observed for Nalgae suggests a highly intermittent multifractal regime characterised by extreme fluctuations in precipitation; however, further analysis is required to determine whether this behaviour reflects a robust physical property or an artefact linked to sampling, the quality of the observations or other factors
Minor comment
Abstract The abstract concludes that the results are relevant to the forecasting and management of extreme hydrometeorological events. This assertion would merit a more detailed explanation. The manuscript does not demonstrate how the multifractal properties identified could be incorporated into forecasting systems or hydrological applications. Either these practical implications should be addressed, or the assertions should be qualified.
Line 76 : Please provide the references corresponding to the statement ‘such underestimation is well documented for lasers disdrometers…’
Figure 2 The statement that the measurements exhibit a "multiplicative, scale-invariant bias" seems too strong. The approximately constant relative bias observed between 1 and 60 min suggests such a behaviour over the analysed scales, but does not demonstrate scale invariance in a general sense. A more cautious wording would be appropriate.
Recommendation: major revision
The manuscript addresses an interesting problem using a valuable observational dataset.
The results are interesting and suggestive, but the interpretation should remain more cautious.
However, its current contribution is weakened by insufficient discussion of novelty, limited assessment of statistical robustness, and an interpretation of phase transitions that is stronger than what is supported by the presented evidence. The manuscript would also benefit from a clearer statement of its scientific objectives and of the new understanding or practical implications expected from the proposed analysis.
Overall, I believe the manuscript addresses an interesting question and is based on a valuable dataset. Addressing the points raised above would considerably improve the robustness of the conclusions.