the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 08 Apr 2026
Maximum Certainty Principle applied to rainfall modelling and regionalisation in Ecuador
Abstract. This study introduces the Maximum Certainty Principle (PCM, from the Spanish “Principio de Certeza Máxima”) as a variational framework for the probabilistic modelling of storm events and its application to the regionalisation of extreme rainfall. First, the PCM is developed and applied in the Metropolitan District of Quito through the Storm Information Model (MIT-Q), which represents the intra-event temporal structure using a truncated exponential formulation derived from the PCM variational functional. Within MIT-Q, event maximum precipitation (PRE) and event duration (DT) are modelled using Weibull and truncated exponential distributions, respectively. Stochastic simulations equivalent to 500 years of rainfall were performed, calibrated against Intensity–Duration–Frequency (IDF) curves from the Quito-Observatory station and validated at four additional stations within an area of approximately 2500 km². Results indicate that extreme rainfall intensities with durations shorter than 2 h are statistically independent of daily intensities, and that the structural separation between PRE and DT improves the representation of short-duration extremes. Second, the structural insights obtained at the local scale are extended to the existing national rainfall regionalisation framework of Ecuador. A scaling factor (𝜑) associated with the potential number of rainfall bursts, representing the dynamics of local winds and the growth of storm cells, is introduced, enabling the derivation of Potential Intensity–Duration–Frequency (IDFP) curves that remain useful under sparse observation network conditions, such as those of the Ecuadorian network. The proposed approach integrates theoretical development, local validation, and regional scaling within a structural probabilistic framework for the analysis of extreme rainfall.
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Status: open (until 09 Jul 2026)
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CC1: 'Comment on egusphere-2026-1317', Diego Escobar, 15 Apr 2026
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AC1: 'Reply on CC1', Franklin Beltrán, 30 Apr 2026
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Please find attached our detailed response to the reviewer’s comments. The document includes point-by-point replies and supporting supplementary data to ensure transparency and traceability of the analyses.
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CC2: 'Reply on AC1', Diego Escobar, 05 May 2026
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Detailed critique of Beltrán's response
This document is a comprehensive reply to Beltrán's "Responses to Reviewer Comments" (April 2026). I conclude that the original deficiencies remain unresolved. The main points are as follows.
Objection to the Abstract and the MIT-Q simulation
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The MIT-Q model has not undergone independent peer review. References are circular (Beltrán 2022, 2023) or inaccessible theses.
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The choice of Weibull and truncated exponential distributions lacks justification; no goodness-of-fit tests against other families (GEV, gamma) are provided.
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Simulating 500 years of extreme events from a single station (Quito-Observatory) and extrapolating to 2500 km² is methodologically unsound, given the strong orographic gradients (>20%).
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Validation is done against official IDF curves, which are themselves statistical approximations, not against raw high-resolution (5-min) rain gauge data. Comparing a model to another approximation is circular.
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Only 4 validation stations are used, giving a density of 0.0016 stations/km² – far too low to capture convective and orographic variability.
Comment 1 – Statistical independence and unverifiable references
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The author replaced "independence" with "weak linear dependence" but still fails to prove independence. An R² ≤ 0.64 does not imply independence.
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The graphs and data cited (Beltrán 1995, Andrade 1997) are not publicly accessible. No raw data are provided in the supplementary material – only MIT-Q simulation outputs. This violates reproducibility.
Comment 2 – The Maximum Certainty Principle (MCP) is equivalent to MaxEnt
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The functional F[f] = ∫[C(t) f(t) – f(t) ln f(t)] dt, when optimised, yields f(t) ∝ exp(C(t)). This is exactly a Gibbs distribution from maximum entropy with an arbitrary prior. No new principle.
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The invocation of Noether's theorem is incorrect: the Euler-Lagrange equation gives a constant, but Noether requires an explicit symmetry – none is identified.
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The choice C(t) = –λ t is ad hoc, leading to a truncated exponential, a classical model (Eagleson, Rodríguez-Iturbe). No derivation from first principles.
Comment 3 – MIT‑Q calibration, validation and spatial contradictions
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Calibration sources: (i) official IDF curves (approximations), (ii) annual event counts from Pourrut & Leiva (1989) – only graphs, no tables, (iii) intra‑event quartiles from Beltrán (1995) – unpublished thesis. None are independently verifiable.
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Validation at 4 stations shows MAPE up to 68% for 360 min at DAC‑Aeropuerto. The author dismisses this as "low intensities", but 68% error is unacceptable.
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The model adjusts to annual isohyets (total annual precipitation). This suffers from equifinality: many different storm-type combinations can give the same annual total. It does not guarantee correct extreme intensities.
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In high mountain areas (>3500 m a.s.l.), long‑duration low‑intensity rainfall dominates, not convective storms. Forcing an annual precipitation match may create false convective events or miss prolonged rainfall contributions. No sensitivity analysis or validation at paramo stations is provided.
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Parameter α₀ ≈ 10 is called "maximum temporal structuring capacity", but when advection is included αₐ varies from 2 to 35, contradicting the interpretation of an intrinsic parameter. Previous references to gravity have been removed without replacement.
Comment 4 – Regionalisation and potential IDF curves
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The derivation of τ remains opaque. Tangency conditions are imposed without justification. Equation 2t₁ e⁻ᵗ¹/(1−e⁻ᵗ¹) = 1−b₁ makes t₁ a function of b₁, yet τ is claimed constant – no reconciliation.
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The "transition storm" is admitted to be unobservable, a conceptual construct without empirical evidence. No testable predictions are offered.
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The scaling factor φ(b)=b−0.237 is purely empirical (regression on 66 stations, R²=0.968). It is not derived from the MCP; the MCP plays no role in the core regionalisation.
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No uncertainty propagation or interpolation method is specified for the maps. The author claims altitude is implicitly included, but interpolation without explicit topography is inadequate for gradients >20%.
Comment 5 – Exaggerated conclusions
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The author toned down some phrases but still claims an "alternative framework". No quantitative comparison with standard methods (L‑moments, ML, Poisson models) is given. No demonstration that potential IDF curves outperform existing ones.
Comment 6 – Reproducibility and cross‑validation
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Only MAPE is added. No cross‑validation, confidence intervals, residual analysis, or tail tests. Primary data remain inaccessible (theses, congress proceedings, old reports without DOI). The promised supplementary material contains only MIT-Q simulation outputs, not raw observations.
Comment 7 – Overall conceptual contribution
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Lack of novelty: MCP = MaxEnt with an arbitrary prior.
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Circular/unverifiable references: most cited works cannot be accessed.
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Insufficient validation: 4 stations, against IDF approximations, 68% MAPE.
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Spatial contradictions: annual total adjustment does not guarantee extreme convective simulation, especially in high mountains.
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Abuse of mathematical terminology: Noether's theorem and variational calculus used as ornament without genuine application.
Conclusion and final recommendation
The manuscript and the author's response do not remedy the fundamental flaws. I recommend rejection. The author should use high-resolution observational data from more stations, validate against raw records, make all data publicly available, justify distribution choices statistically, and address orographic effects explicitly.
Sincerely,
Diego Escobar-González-
AC3: 'Reply on CC2', Franklin Beltrán, 15 May 2026
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Please find attached our detailed response to the reviewer’s comments. The document includes point-by-point replies and supporting supplementary data to ensure transparency and traceability of the analyses. The supplementary files supporting this response are also being submitted separately as AC4.
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AC4: 'Reply on CC2 – Supplementary material for AC3', Franklin Beltrán, 15 May 2026
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Please find attached the supplementary material associated with our response AC3. The files include scanned historical documents, technical reports, theses and supporting bibliographic sources used in the methodological development of the PCM/MIT-Q scheme.
This material is provided to strengthen the traceability and verifiability of historical sources that are not currently available through DOI-based repositories or open digital platforms. The supplementary files should be read in conjunction with the point-by-point response submitted as AC3.
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AC5: 'Reply on CC2- MIT-Q peer review clarification', Franklin Beltrán, 15 May 2026
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Regarding the peer-review status of the initial MIT-Q scheme, the following clarification is provided.
An initial version of the MIT-Q scheme, together with its initial calibration, was presented in the paper “Principio de Certeza Máxima (ℵmax)”, published in Revista Politécnica (Vol. 52, No. 2), a scientific journal with an editorial and peer-review process. The paper was received on 26/10/2022, accepted on 24/08/2023, and published online on 14/11/2023, with DOI: 10.33333/rp.vol52n2.05.
That paper presents the variational formulation, the theoretical foundations of the proposed framework, and an initial calibration of the MIT-Q scheme for a spatial domain of 1600 km2.
The present manuscript extends that initial application by presenting, for the first time, the calibration and evaluation of the MIT-Q scheme over a 2500 km2 domain corresponding to the Metropolitan District of Quito. Therefore, the current contribution should be understood as a new spatial extension and broader hydrological application of an initial MIT-Q scheme whose first calibration had already been presented in a peer-reviewed scientific publication.
This clarification is submitted as a specific complement to AC3, in order to address the observation concerning the peer-review status of the initial MIT-Q scheme, raised in Comment 2.1 of CC2.
Citation: https://doi.org/10.5194/egusphere-2026-1317-AC5
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CC2: 'Reply on AC1', Diego Escobar, 05 May 2026
reply
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AC2: 'Reply on CC1', Franklin Beltrán, 30 Apr 2026
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Supplementary dataset supporting the analyses presented in the response to reviewer comments, including calibration, validation, and correlation analyses of the MIT-Q model. The dataset includes a README sheet describing its structure and contents to ensure traceability.
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AC1: 'Reply on CC1', Franklin Beltrán, 30 Apr 2026
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Data sets
Data for: "Maximum Certainty Principle applied to rainfall modelling and regionalisation in Ecuador" Franklin Aparicio Beltrán Vega and Jhoan Alexander Beltrán Valarezo https://zenodo.org/records/18916818
Video supplement
Modelo de Información de Tormentas MIT-Q (MIT-Q) Jhoan Alexander Beltrán Valarezo https://www.idd-research.org/modelo-de-informacion-de-tormentas-mit-q
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I have read with interest your preprint entitled “Maximum Certainty Principle applied to rainfall modelling and regionalisation in Ecuador” (EGUSPHERE-2026-1317), and I must state the following:
Abstract and Introduction
The abstract claims that statistical independence has been demonstrated between short-duration rainfall intensities (less than 2 hours) and daily intensities, based on a correlation coefficient of R2≤0.64. This value alone does not imply independence; it confuses basic statistical concepts and exaggerates the findings. Furthermore, the introduction invokes Noether's theorem and symmetries to justify a "probabilistic invariant", but the symmetry is never identified, so the analogy is forced.
Foundations of the Maximum Certainty Principle (sections 2.1 to 2.2)
The variational formulation of the PCM does not constitute a new principle, but rather a reformulation of the Maximum Entropy Principle (MaxEnt) with an arbitrary prior. The central functional ℵmax where C(t) is vaguely defined as a "knowledge potential", directly leads to the solution fT(t). Mathematically, this is equivalent to standard entropy maximization considering an exponential distribution, as described in Kapur (1989). More seriously, explicit Lagrange multipliers are not derived, nor is the Euler-Lagrange equation solved rigorously; it is mentioned in a circular reference that this is done in Beltrán (2023), but no demonstration is provided. The resulting truncated exponential distribution for the intra-event structure is a classic model in hydrology (Eagleson, 1978; Rodríguez-Iturbe et al., 1987), used for decades. Renaming it as the "Maximum Certainty Principle" does not represent a real methodological advance. Moreover, assuming that C(t) is monotonic and independent of elevation is physically unsustainable in the Metropolitan District of Quito (DMQ), where rainfall intensity exhibits strong altitudinal gradients between 2600 and 4555 m a.s.l., as shown in [missing reference].
MIT-Q Model (sections 2.3 to 2.3.4)
The model is calibrated with daily data from a single station (Quito‑Observatory) for the period 1916‑1992, completely ignoring modern high‑resolution rain gauge networks (5‑minute data) that have been operating in the DMQ for at least 20 years. In a context of climate change, using such old records violates the stationarity assumption and fails to adequately capture recent extreme events, especially short‑duration ones. The author does not explain how sub‑hourly intensities are derived from daily data, which is a serious limitation. A descriptive analysis of the pluviometric information from the Quito‑Observatory station should have been performed, indicating whether the records are sub‑hourly, hourly, daily, or band data. Nothing is explained about the data source; it only says "the century‑old Quito‑Observatory station", here and in all of Beltrán's circular references.
Furthermore, the model includes a dimensionless parameter calibrated to a value close to 10 and vaguely related to the acceleration of gravity (g). Note that in Beltrán (2023) it is stated that " la gravedad (g) induce información que obliga la conformación de patrones aleatorios preferentemente precoces", without any physical derivation justifying this relationship. The explanation of "upward pulses suppressed by gravity" is a metaphor or pseudoscience. Later, this same parameter varies between 2 and 35 depending on the station, contradicting the supposed universality of the gravitational constant.
Model validation is presented only through visual comparisons of IDF curves (Figure 5), without quantitative error metrics that show model skill, such as RMSE, BIAS, confidence intervals, or statistical tests. Validation based solely on graphical inspection is insufficient for a work that claims to provide a new methodology for extreme rainfall regionalisation.
Regionalisation and Potential IDF curves (sections 2.4 to 3.1)
The derivation of the key equation relating the parameter τ to the empirical coefficient bb of the INAMHI IDF curves is presented opaquely, with algebraic steps omitted. It is not clear how the expression 1−b1 is reached nor why τ is assumed constant for all durations. The introduction of the "transition storm" appears to be a mathematical artifice to join two duration regimes, without observational evidence that storms with that property exist.
The scaling factor ϕ(b)=b−0.237 is obtained through an empirical regression (Figure 8a), but goodness‑of‑fit measures are not reported nor is estimation uncertainty discussed. Most worryingly, this regression is not derived from the Maximum Certainty Principle; it is a purely empirical correction that has nothing to do with the variational functional presented in the first part of the paper.
The scaling factor maps (Figure 9) are presented as deterministic results, but no uncertainty propagation from station parameters to map cells is included. Nor is the spatial interpolation method specified. In a region with strong orographic gradients (exceeding 20%), ignoring altitude and topography is a serious omission. Note: precipitation is not random.
Discussion and Conclusions
The conclusions exaggerate the scope of the work. It is claimed that the PCM "modifies the inferential interpretation of extreme event modelling", but this is not demonstrated with concrete examples or comparisons against established methods (L‑moments, maximum likelihood, two‑state Poisson models). The potential IDF curves are presented as an operational contribution, but they lack independent validation with dense high‑resolution networks. There is no analysis quantifying the improvement over IDF curves generated by EPMAPS or INAMHI.
Appendix and Reproducibility
The calibration and validation are insufficient for the regionalization the author seeks to justify. The Storm Information Model (MIT‑Q) is calibrated at a single station (Quito‑Observatory, 1916‑1992) and validated only at four additional stations over approximately 2500 km². This control point density is extremely low to capture convective and mesoscale processes, as well as the dominant orographic effects in the Andes. The approach lacks quantitative cross‑validation metrics and does not evaluate model skill using statistics such as MAE, RMSE, NSE, or MAPE, nor does it demonstrate superiority for short‑duration (<2 h) extreme events, where tail behavior is critical.
Finally
The work mixes a variational theory that provides no real novelty (equivalent to maximum entropy) with a storm model calibrated using obsolete data and an empirical regionalization that is not derived from the proposed principle.
Att. diego.p.escobar.g@gmail.com