Developed by Francesco Cappelli, Salvatore Grimaldi, Andrea Petroselli, and Emanuele Santinami of Tuscia University in Viterbo, Italy, Virtual Rain v1.0 does not invent new rainfall mathematics. Instead, its novelty lies in integration. For the first time, two previously separate and validated methodologies, the CoSMoS-2s daily rainfall generator and the Multifractal Random Cascade disaggregation model, are available together in a single, reproducible software environment, implemented through Python and R routines and mirrored on an interactive web platform that requires no programming skills.

The problem the toolkit addresses is structural. Hydrologists typically have access to long daily precipitation time series, often spanning several decades, plus fragmentary observations of short, intense storms used to build intensity-duration-frequency curves, the workhorse charts of infrastructure design. What they lack is continuous sub-daily data, which stochastic precipitation models need in abundance to capture the multiscale structure of rainfall. Existing generators and disaggregation techniques abound in the literature, but choosing and wiring together an appropriate combination has remained a genuine obstacle for practitioners, particularly in data-sparse regions.

Virtual Rain’s first module tackles daily simulation using CoSMoS-2s, a two-state stochastic framework that explicitly separates rainfall occurrence from rainfall intensity. In the first state, the observed record is transformed into a binary wet-or-dry process, with seasonally varying occurrence probabilities and autocorrelation structures mapped into a Gaussian autoregressive model and converted back through thresholds. In the second state, wet-day rainfall amounts are fitted with parametric distributions, including Gamma, Generalized Gamma, Burr, and Weibull families, while their persistence is reproduced through parametric or empirical autocorrelation functions. The two components are then recombined, allowing the synthetic series to preserve intermittency, seasonal cycles, marginal distributions, and temporal dependence simultaneously.

The second module performs the disaggregation. It assumes that rainfall intensity evolves as a multiplicative cascade, a multifractal process in which the mean intensity at each timescale is redistributed through random weights following a beta-lognormal distribution. Three parameters govern the cascade: one controlling intermittency and the share of dry intervals, one governing the intensity fluctuations, and an outer-scale parameter defining the range of the multifractal behaviour. Crucially, rather than demanding high-resolution observations for calibration, the model derives these parameters from IDF curves, matching theoretical multifractal intensity-duration relationships to the target curves through numerical optimization. Because IDF information is widely available even where fine-scale gauges are not, this dramatically broadens the framework’s reach.

The toolkit guides users through a structured workflow. On the simulation side, routines handle exploratory plotting, fitting of seven candidate marginal distributions using the Nelder-Mead algorithm, estimation of autocorrelation structures for both the binary and continuous processes, ensemble generation of synthetic series, and diagnostic comparison of observed versus simulated statistics, from dry-day probability to extreme percentiles. On the disaggregation side, routines extract annual maxima, fit generalized extreme value distributions by maximum likelihood, probability weighted moments, or L-moments, calibrate the cascade parameters against IDF targets, and finally break daily totals down to resolutions as fine as one minute, with an option to preserve daily rainfall volumes exactly.

To demonstrate the software, the authors applied the complete workflow to a rain gauge in the Arno River Basin in Tuscany, one of Italy’s longest river systems, using a 20-year daily record from a network of more than 70 gauges with continuous 15-minute observations. The Gamma distribution emerged as the best overall fit for wet-day intensities, and an ensemble of 50 synthetic series successfully reproduced the observed dry frequency, seasonal cycle, autocorrelation, and ranked annual maxima, with only a slight overestimation in the upper tail that the authors attribute to the limited sample of wet days in a record that is nearly 70 percent dry. The imposed IDF scaling exponent, estimated at 0.251 from the observed sub-daily data, and the dry-day frequency were both recovered exactly in the disaggregated output, confirming that the integrated pipeline preserves the statistical properties it is calibrated to reproduce.

The authors are careful about what the case study does and does not show. Because the same observational record serves for calibration and diagnostics, the comparison verifies the software implementation rather than validating predictive performance on independent data, and the methodological validation of the underlying models rests on earlier publications. They also flag honest limitations: parameter uncertainty is not yet propagated through the full simulation-disaggregation chain, some combinations of cascade parameters can fit equally well, and short or incomplete records can produce unstable estimates. Diagnostic outputs, they stress, should be treated as an integral part of model configuration, not an afterthought.

What makes Virtual Rain potentially transformative is accessibility. Beyond the Python and R routines, an interactive web platform walks users step by step from data upload through seasonal partitioning, distribution fitting, ensemble diagnostics, cascade calibration, and final disaggregation, with both interfaces implementing identical formulations and parameterizations. The software is archived on Zenodo under a GPL-3.0 licence, and the Tuscan rainfall data are openly available under a Creative Commons licence. For engineers designing drainage systems, researchers assessing flood risk under climate change, or teachers demonstrating stochastic hydrology, the toolkit lowers a barrier that has long separated sophisticated rainfall theory from everyday practice, turning decades of ordinary daily rain gauges into a window on the violent, minute-scale storms that matter most.

Subject of Research: Stochastic simulation and temporal disaggregation of rainfall time series for hydrological modelling

Article Title: Virtual Rain v1.0: a unified toolkit for high-resolution rainfall simulation and disaggregation

Article References: Cappelli, F., Grimaldi, S., Petroselli, A., & Santinami, E. (2026). Virtual Rain v1.0: a unified toolkit for high-resolution rainfall simulation and disaggregation. Geoscientific Model Development, 19(19), 9489-9518. https://doi.org/10.5194/gmd-19-9489-2026

Image Credits: AI Generated

DOI: 10.5194/gmd-19-9489-2026

Keywords: rainfall simulation, stochastic modelling, disaggregation, CoSMoS-2s, multifractal random cascade, IDF curves, flood risk, hydrology, open-source software, time series generation, Geoscientific Model Development, Tuscia University

The research, led by Aaron Jerome Rampersad of the University of Canterbury with Christianne Marie-Claire Faith Zakour of the Loss and Damage Youth Coalition, addresses a problem that has haunted Caribbean hydrology for decades. Rainfall networks across the region are sparse, records are riddled with gaps, and conventional methods for building design rainfall curves quietly assume data that simply do not exist. The stakes are not abstract. Hurricane Ivan in 2004 damaged or destroyed approximately 89 percent of Grenada’s housing stock, inflicting losses near 900 million US dollars, roughly twice the national GDP. Hurricane Beryl in 2024 caused an estimated 218 million dollars in damage and triggered parametric insurance payouts of 55.6 million dollars. Designing infrastructure against the wrong rainfall statistics has direct, measurable consequences.

The team assembled an archive of 28 rain gauges drawing on records from Grenada’s National Water and Sewerage Authority and the Grenada Airports Authority. Before any analysis, the raw material was daunting: 7,671 observed station-days from the water authority network alongside 14,885 daily values from the airport, with most non-airport stations suffering gaps ranging from isolated days to entire missing years. Only Point Salines, with an approximately continuous 40-year record, approached the completeness that standard frequency analysis assumes. For many stations, the eventual curated dataset guaranteed a minimum of 12 years of usable records, a thin foundation for estimating rainfall quantities associated with 50- or 100-year return periods.

The workflow begins with a diagnostic innovation. Rather than relying on the classical semivariogram, the geostatistical workhorse that bins station pairs by separation distance, the authors computed site-specific Pearson correlations directly between every pair of stations. This approach, adapted from techniques used to study non-stationary spatial correlation in earthquake ground motions, sidesteps a known weakness: with so few stations, lag-bin averaging obscures the very structure the analysis is meant to reveal. The verdict was clear. Daily rainfall dependence in Grenada is governed primarily by how far apart two stations are, with elevation acting as a secondary influence whose effect shifts with the seasons. March, one of the driest months, showed strikingly coherent spatial rainfall, while June, as the Intertropical Convergence Zone migrates northward, produced far more scattered behavior.

Those diagnostics fed directly into the gap-filling stage. The team fitted spatial correlation models to the daily rainfall field using Bayesian inference, implemented in Python with the NumPyro library and the No-U-Turn Sampler, treating model parameters as probability distributions rather than fixed numbers. Ordinary kriging driven by these Bayesian-inferred models then reconstructed missing daily values, and it outperformed a full bench of competitors including inverse distance weighting, radial basis functions, and Gaussian process regression. Adding elevation dissimilarity as a covariate delivered marginal but consistent gains. Strict quality control followed: imputations were only retained when at least six donor stations contributed and the kriging prediction variance stayed below 25 percent of the marginal daily variance, lifting every station’s completeness above 70 percent.

With the reconstructed dataset in hand, the researchers turned to extreme value theory. Instead of the Gumbel distribution used in earlier Grenadian studies, which effectively fixes the shape parameter at zero and can underestimate rare rainfall quantiles, they fitted the full generalized extreme value distribution to annual maxima and the generalized Pareto distribution to peaks over threshold, the latter using declustering to ensure that exceedances separated by less than 24 hours counted only once. Because most stations contributed just 12 years of annual maxima, weakly informative priors were specified through an empirical Bayes-style strategy informed by preliminary analytical fits, stabilizing inference on the tail-shape parameter that controls how heavy the rainfall distribution’s upper end truly is. Markov chain Monte Carlo sampling, with four chains and extended warm-up, produced full posterior distributions for every parameter.

The physical signals that emerged are plausible for a mountainous Caribbean island. The shape parameter correlated moderately with elevation, at 0.25 for the generalized extreme value model and 0.34 for the generalized Pareto model, consistent with orographic enhancement steepening the upper tail of the rainfall distribution where moisture-laden trade winds are forced over the central highlands. Interpolating the posterior return levels across the island required choosing among kriging variants, and intrinsic collocated cokriging with a residual correlogram, which exploits elevation as a secondary variable, won on leave-one-out cross-validation. Five poorly constrained stations with only about four years of reliable data were excluded after sensitivity testing showed they destabilized the fitted spatial structures. The final maps show the highest predicted extremes in the island’s northeast, broadly matching Grenada’s known climatic zoning, though accompanied by appropriately large uncertainty estimates.

The study is unusually candid about its own limitations. A sensitivity analysis traced every annual maximum and threshold exceedance back to its source, classifying each as observed or imputed, and then refitted the models using observed extremes only. Where imputed values dominated a station’s extreme sample, return levels shifted dramatically, with differences approaching 80 percent for some generalized Pareto estimates and exceeding 50 percent for some extreme value estimates; at the 25-year return period, most stations stayed within roughly 25 percent. The authors attribute this partly to the smoothing inherent in kriging, which produces conditional-mean predictions rather than stochastic realizations and can dampen localized extremes. They suggest that future work propagate imputation uncertainty directly, through empirical Bayesian kriging, multiple conditional realizations, or hierarchical models treating missing rainfall as latent quantities.

What elevates the work beyond a single-country case study is its transferability. The complete workflow, from site-specific correlation diagnostics through Bayesian model fitting to island-wide interpolation, is publicly available through GitHub and Zenodo repositories, and the underlying Grenada Daily Rainfall Database has been released on Zenodo. The authors also outline an engineering validation path, proposing two-dimensional flood simulations in flood-prone catchments such as St. John’s and Great River to test whether the estimated rainfall fields produce physically reasonable inundation. For small island developing states facing intensifying hurricanes and rising adaptation costs, the message is straightforward: with Bayesian methods, even a fragmented, decades-old network of rain gauges can yield defensible, spatially explicit design rainfall, provided the uncertainties are confronted rather than hidden.

Subject of Research: Bayesian estimation of multi-station rainfall intensity-duration-frequency curves and extreme daily rainfall mapping in data-limited island settings, applied to Grenada

Article Title: A Bayesian workflow for multi-station IDF curve development in data-limited island settings: application to Grenada

Article References: Rampersad, A. J., & Zakour, C. M.-C. F. (2026). A Bayesian workflow for multi-station IDF curve development in data-limited island settings: application to Grenada. Theoretical and Applied Climatology, 157(10), Article 633. https://doi.org/10.1007/s00704-026-06558-4

Image Credits: AI Generated

DOI: 10.1007/s00704-026-06558-4

Keywords: IDF curves, extreme rainfall, Bayesian inference, Grenada, kriging imputation, generalized extreme value distribution, generalized Pareto distribution, spatial correlation, small island states, geostatistics, return levels, climate risk