| 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.
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