Most existing schemes inside NEMO treat sunlight as a single downward stream that simply fades with depth, resolved over only two to five broad wavebands. The new module, adapted from the model developed for the MITgcm-DARWIN configuration, does something fundamentally different. It separates the light field into three distinct streams: a direct downward stream that has not yet been scattered, a scattered downward stream produced by light deflected forward by particles and water molecules, and an upward stream of backscattered light heading back toward the surface. Each stream is solved independently for every wavelength, and the model runs across 33 wavebands spanning 250 to 4000 nanometres, with a fine 25-nanometre resolution in the visible range where biology matters most.

The technical heart of the module is a set of coupled equations describing how each stream is attenuated by absorption, forward scattering, and backscattering as it propagates through the water column. These processes are governed by the inherent optical properties of seawater, which are determined by what is actually in the water: pure seawater itself, phytoplankton, detritic non-algal particles, and coloured dissolved organic matter, known as CDOM. In the coupled configuration, a biogeochemical model supplies the concentrations of these optically active constituents, and the module converts them into spectral absorption and scattering coefficients. CDOM, which absorbs strongly at short wavelengths, is handled with an exponential decay law referenced at 412 nanometres, while phytoplankton optics are computed from three functional types relevant to the Black Sea: diatoms, small flagellates, and large flagellates.

What makes this development genuinely powerful is the feedback loop. The simulated irradiances are not just diagnostic outputs; they are fed back into the physics and the biology. The total scalar irradiance budget becomes the source term in the temperature equation of NEMO, replacing the crude attenuation schemes of the past, and the visible-range scalar irradiance available to phytoplankton drives primary production in the coupled ecosystem model. This two-way coupling maintains physical consistency: the same light that heats the water is the light that plankton use to grow. The researchers calibrated a tuning factor, set to 0.85 in the Black Sea configuration, that accounts for the small fraction of absorbed energy used by photosynthesis and particle degradation rather than heating.

The test case is the Black Sea, a basin whose biogeochemistry is simulated operationally by the Copernicus Marine Service using the NEMO-BAMHBI framework. BAMHBI, the BiogeochemicAl Model for Hypoxic and Benthic Influenced areas, explicitly represents the anoxic deep layer of the Black Sea and the full marine foodweb from bacteria to mesozooplankton. Previously it relied on a simple three-band optics scheme in which a single downward light stream was attenuated following Beer’s law. Substituting that scheme with the new three-stream model required careful calibration, which the team performed using more than a decade of data from Biogeochemical Argo floats drifting through the basin, measuring chlorophyll, particle backscattering, CDOM fluorescence, and downwelling irradiance at 380, 412 and 490 nanometres.

The surface boundary conditions come from an unexpected direction: the sky. The team used the MAR regional atmospheric model, running with the ECMWF ecRad radiation scheme and a high-resolution gas-optics configuration, to simulate spectral shortwave fluxes over the Black Sea in the same 33 wavebands used by the ocean module. MAR separates direct and diffuse irradiance, exactly the split the ocean model needs, and validation against ground observations showed correlations above 0.9 with low bias. Importantly, because sea surface reflectance is computed as a ratio of upwelling to downwelling irradiance, any residual bias in the atmospheric forcing largely cancels out, making the reflectance output robust.

The validation results are encouraging. Simulated downwelling irradiances at 412 and 490 nanometres agree very well with BGC-Argo profiles, with regression slopes close to one, although the model absorbs slightly too much light near the surface at shorter wavelengths where CDOM dominates. Sea surface temperature simulations remain consistent with observations, with correlations above 0.99, and the mixed layer depth, thermocline, and cold intermediate layer of the basin are preserved. Surface chlorophyll improves modestly, with the root mean square error dropping by 0.16 milligrams per cubic metre, though the model still tends to overestimate winter and spring blooms. The most striking gap appears during the May-to-July coccolithophore bloom, when the model underestimates reflectance at 490 nanometres because it does not capture the intense scattering from these calcifying plankton.

Perhaps the most transformative capability is the direct simulation of sea surface reflectance, the quantity that satellites actually measure. Rather than comparing modelled chlorophyll with satellite-derived chlorophyll through uncertain inversion algorithms, the model now produces reflectance fields that can be compared directly with remote-sensing data. The team went further and applied the same band-ratio inversion algorithm used operationally for Black Sea satellite products to their own simulated reflectance, producing a reflectance-derived chlorophyll estimate. This estimate agreed better with the satellite product than the dynamically simulated chlorophyll for most of the year, with a substantially lower error, particularly during the early spring bloom. The exercise also revealed a subtlety: the coccolithophore bloom visible in satellite reflectance does not appear in the satellite chlorophyll signal, because increases in both blue and green reflectance cancel out in the band ratio, suggesting the satellite product itself may be biased during such blooms.

The module also ships with a stochastic mode designed to confront one of the field’s thorniest problems: uncertainty in optical properties. Using first-order autoregressive processes with roughly 75-kilometre spatial correlation and one-month temporal correlation, the model perturbs absorption and scattering coefficients with standard deviations of up to 50 percent, following log-normal distributions consistent with the statistics of optical variables. A 50-member ensemble compared against in situ reflectance measurements from the AERONET-OC Galata tower off the Romanian coast showed that observations generally fell within the ensemble spread, with uncertainty widening during bloom periods. This quantification of model uncertainty is precisely what is needed for future data assimilation of reflectance, a step that could revolutionise how satellite ocean colour constrains ecosystem forecasts.

There are trade-offs. Running the radiative transfer model at every time step across 33 wavebands roughly doubles the computation time compared with the simple optics scheme, a cost that matters for long-term or global simulations, though it can be reduced by lowering spectral resolution or simplifying the scattering treatment. The authors also note limitations: the assumption of no bottom reflection restricts some coastal applications, phytoplankton optical properties were derived partly from Mediterranean observations, and suspended minerals near river mouths are not yet explicitly represented. Still, the implications reach well beyond the Black Sea. With hyperspectral satellite missions such as NASA’s PACE and the Italian Space Agency’s PRISMA now delivering reflectance data at unprecedented spectral resolution, models capable of resolving light propagation in fine wavebands are becoming essential. By making sophisticated radiative transfer a pluggable component of NEMO, this work opens a path toward ocean models that speak the same optical language as the satellites watching them, bringing simulation and observation into direct, quantitative conversation.

Subject of Research: Three-stream spectral radiative transfer modelling coupled with biogeochemical simulation of the Black Sea in the NEMO ocean modelling framework

Article Title: Three-stream modelling of radiative transfer for the simulation of Black Sea biogeochemistry in a NEMO framework

Article References: Macé, L., Vandenbulcke, L., Brankart, J.-M., Grailet, J.-F., Brasseur, P., & Grégoire, M. (2026). Three-stream modelling of radiative transfer for the simulation of Black Sea biogeochemistry in a NEMO framework. Geoscientific Model Development, 19(18), 9077-9102. https://doi.org/10.5194/gmd-19-9077-2026

Image Credits: AI Generated

DOI: 10.5194/gmd-19-9077-2026

Keywords: radiative transfer, NEMO, Black Sea, biogeochemical modelling, ocean optics, sea surface reflectance, phytoplankton, CDOM, BGC-Argo, remote sensing, Copernicus Marine Service, stochastic modelling

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 tool, called EWF 2.0, was developed by Jaromir Sant of the University of Turin and the Collegio Carlo Alberto, together with Paul A. Jenkins, Jere Koskela and Dario Spanò of the University of Warwick and Newcastle University. It extends an earlier line of work known as the EWF algorithms, which allow researchers to draw exact samples from the Wright–Fisher diffusion — that is, to generate simulated allele frequency trajectories whose statistical properties match the theoretical model perfectly, without the approximation errors that plague standard numerical schemes. The crucial innovation in version 2.0 is that the population size is no longer required to be constant. Instead, the demographic history can change over time, which mathematically turns the drift coefficient of the diffusion into a time-inhomogeneous one: a process whose random dynamics depend explicitly on when you look at it.

To appreciate why this matters, it helps to understand what exact simulation means in this context. The Wright–Fisher diffusion is a stochastic differential equation, and the most common way to simulate such equations is to discretise time into small steps and approximate the motion between them. These Euler-type schemes are easy to implement but introduce bias: the simulated trajectories are not true draws from the model, and the bias can be difficult to quantify, particularly for quantities that depend on the fine structure of the path rather than just its endpoints. Exact algorithms sidestep this problem entirely. Using clever probabilistic constructions — often based on rejection sampling ideas in which a candidate path is proposed and then accepted or rejected according to criteria that guarantee the output has exactly the right distribution — they produce trajectories that are statistically indistinguishable from the real thing, whatever the intended application.

The difficulty with time-varying demography is that these rejection-based constructions were designed for diffusions with fixed coefficients. When the population size changes, the drift term of the allele frequency dynamics acquires a time-dependent factor, and the bounds and transformations that make the earlier algorithms valid no longer hold. Previous versions of EWF could handle time-varying mutation and selection in certain settings, but the demographic component had to remain static. Since virtually every dataset from natural populations — humans, livestock, pathogens, crop species — carries the imprint of a fluctuating census history, this limitation was a serious obstacle for anyone hoping to use exact simulation in realistic inference pipelines.

EWF 2.0 removes that obstacle. According to the authors, the algorithm accommodates a specified demographic history alongside mutation parameters, a selection function and a set of sampling times, and from those inputs it generates exact draws from the law of the corresponding Wright–Fisher diffusion. Perhaps more strikingly, it can also simulate diffusion bridges: trajectories that are conditioned to start at one allele frequency and end at another. Bridge simulation is a notoriously hard problem, because the conditioning fundamentally alters the dynamics of the process, and the authors note that existing methods cannot handle time-varying mutation and selection rates in this setting. Bridges are precisely what many statistical applications need, since inferring past evolutionary forces often amounts to asking which hidden paths most plausibly connect observed genetic data points collected at different times.

The applications of such a tool extend across modern population genetics. Ancient DNA studies, for example, now routinely provide snapshots of allele frequencies at multiple time points separated by hundreds or thousands of generations. Reconstructing the trajectories between those snapshots — and quantifying the uncertainty around them — requires simulating bridges under a model that includes the demographic history inferred from archaeological and genomic evidence. Similarly, forward-time simulations used to benchmark inference methods, to test the power of genome-wide association designs, or to explore how selection responds to environmental change all benefit from trajectories that are exact rather than approximate. Any bias in the simulator propagates into the conclusions drawn from it, so the guarantee of exactness is not a mathematical luxury but a practical safeguard.

The authors validated their implementation using distributional tests, including Kolmogorov–Smirnov tests and quantile-quantile plots, which compare the empirical distribution of simulated quantities against theoretical expectations. Agreement in these tests provides strong evidence that the algorithm samples from the intended law, since even subtle biases in a simulation scheme tend to show up as systematic deviations in such comparisons. Equally important from a practical standpoint is the performance result: despite its greater generality, EWF 2.0 retains the same runtime as previous versions of the algorithm. Adding the flexibility of time-varying demography therefore comes at no computational penalty, which the authors highlight as ensuring efficiency and scalability for large-scale simulation studies.

The software is written up as an open-access methods paper and the code is freely available on GitHub, lowering the barrier for other groups to adopt it. In a field where simulation underpins everything from method validation to likelihood-free inference, a tool that combines exactness, demographic realism and competitive speed is likely to find rapid uptake. The work was supported by funding from the Italian Ministry of University and Research through a PRIN 2022 grant financed by the European Union’s Next Generation EU programme, reflecting the increasingly international and computationally oriented character of statistical genetics research.

Looking at the broader picture, the advance illustrates a familiar pattern in computational science: the gap between the models researchers want to use and the models their tools can actually handle. The Wright–Fisher diffusion with time-varying demography has been part of the theoretical literature for decades, and approximate simulators have long been able to produce plausible-looking trajectories under it. What was missing was the guarantee — the assurance that a simulated history is a genuine draw from the specified model rather than an artefact of numerical approximation. By extending exact simulation into the time-inhomogeneous regime, EWF 2.0 brings a foundational class of population genetic models fully within reach of rigorous simulation-based inference, and it does so without asking practitioners to sacrifice speed. For studies probing how genes rose and fell across the shifting demographic tides of the past, that combination may prove hard to resist.

Subject of Research: Exact simulation of allele frequency trajectories under the Wright–Fisher diffusion with time-varying demography

Article Title: EWF 2.0: exact sampling of allele trajectories using the Wright–Fisher diffusion with time-varying demography

Article References: Sant, J., Jenkins, P. A., Koskela, J., & Spanò, D. (2026). EWF 2.0: exact sampling of allele trajectories using the Wright–Fisher diffusion with time-varying demography. BMC Bioinformatics. https://doi.org/10.1186/s12859-026-06576-z

Image Credits: AI Generated

DOI: 10.1186/s12859-026-06576-z

Keywords: Wright–Fisher diffusion, exact simulation, population genetics, allele frequency, demography, genetic drift, selection, mutation, diffusion bridge, computational biology, stochastic modelling, BMC Bioinformatics