The revolution of numerical methods for geosciences

Research Article published on 10 October 2025 , Updated on 10 October 2025

Numerical methods are central to geosciences, giving scientists powerful tools that can model and understand the complex processes at work on planet Earth. These tools are currently experiencing spectacular breakthroughs.

Whether simulating climate change or predicting ocean dynamics, numerical methods model phenomena that are impossible to observe directly or reproduce in the laboratory. They are based on mathematical models translated into algorithms, executed using ever greater processing capacities. Long limited by computing power and data availability, they are now benefiting from the rapid developments in high performance computing, machine learning and space-based observations. This is paving the way for more realistic and faster simulations that are better adapted to the uncertainty inherent in natural systems. These techniques are no longer confined to describing the past or present; they test future scenarios giving scientists and decision-makers new ways of anticipating environmental changes and informing choices for the future.

The (necessary?) increase in resolution

The search for precision leads to major challenges in the world of research, particularly in terms of climate model resolution. Finer grids are needed to incorporate highly localised phenomena, improve the understanding of certain biophysical processes and produce more robust simulations. However, this also requires an increase in computing power and the amount of data, which is accompanied by a cost in terms of both time and energy. Anne Cozic, a research engineer in the Calculations team at the Laboratory for Climate and Environmental Sciences (LSCE – Univ Paris-Saclay/French National Centre for Scientific Research (CNRS)/French Alternative Energies and Atomic Energy Commission (CEA)/UVSQ), remembers the initial increases in resolution.

"At the time, we were parallelizing codes," she says, describing the method that divides tasks across multiple processors and runs them simultaneously, providing a benefit in terms of computing time, resolution, complexity and the amount of data processed. "This calls for an entirely different coding system to structure the work and manage data flow, but it has allowed us to increase resolution while maintaining reasonable computing times," explains Anne Cozic. Generally, each processor receives one block of the grid, and the set of processors works more quickly than a single processor.

Terre découpée en mailles

However, this grid, which divides the Earth into longitude and latitude, is a problem in itself, as it is not regular.  The meshes, more or less square at the equator, become distorted at the poles and hamper the speed of calculations, making the parallelization of codes less efficient. "A few years ago, a team developed a new, unstructured grid, with icosahedron sections instead of rectangles," reveals the engineer.

This approach, which is similar to the shape of a football is uniform across the entire surface of the globe and consequently, easier to divide across several processors. "This was useful for a study by Zoé Lloret (during her thesis), for quantifying national CO2 emissions budgets," Anne Cozic points out. Using this new grid, scientists have been able to parallelize codes and achieve a finer resolution. This is essential for informing adaptation strategies for national budgets, in cases where the regional divisions in the models are not sufficient.

The GPU generation

"The next generation will be very different and achieve a much higher resolution," confirms the LSCE scientist.  "We are right in the middle of this in-between phase." This next generation of processors are Graphics Processing Units (GPUs), originally developed for use in video gaming. These technologies, designed to handle the intensive and essential mathematical calculations for graphics rendering and image processing, have proven to be particularly valuable in geosciences. The new generation of processors use them to process more data more quickly before compiling these data on a central processing unit (CPU) capable of handling the simulation's more complex tasks.

Graphics Processing Unit

This means that all scientists working with codes designed for CPUs need to port their code so that it can be read and executed on GPUs. "Some working groups are busy creating a tool that would operate before the data are compiled [GPU to CPU] to re-write the code and make life easier for all users," explains Anne Cozic. She believes that this is one of the major benefits of being part of a research federation (Institut Pierre-Simon Laplace) which facilitates inter-laboratory working groups and develops shared tools. "It's not easy to adapt and today, that's where the challenge lies," she stresses. "Users are going to have to adjust to these codes for GPU. And the more complex these new codes, the greater the risk of error."

The Earth sciences community is working on similar tools, such as the supercomputers on which climate models run. The decision by the Institute for Development and Resources in Intensive Scientific Computing (Idris – Univ. Paris-Saclay/CNRS) to switch to GPUs means that supercomputer users must port their code accordingly. Otherwise, research teams would only be able to use a small part of it. Nevertheless, Anne Cozic maintains that it is vital to question the actual needs of the project and consider the energy cost. "Our atmospheric chemistry models include vast numbers of chemical species and are extremely complex - increased resolution is not a priority. Everyone has to find the balance between power, computing time and energy cost."

Parametrizations and their bias

Numerical methods offer other solutions for integrating processes that climate models cannot explicitly resolve. These include cloud formation where the spatial scale is much finer than the model grid of around 100 kilometres. Scientists then rely on parametrization: simplifications used to approximate the effects of these phenomena across the model's variables. These involve choices and create tiny differences in each model, which become substantial by the end of the simulation, particularly in terms of the Earth's energy balance. Soulivanh Thao, a researcher in the Extremes: statistics, impacts and regionalisation (Estimr) team at LSCE, explains that "certain models, especially impact and hydrological models, are very sensitive to threshold effects and bias correction then becomes critical."

In order to overcome this bias and obtain robust projections, models are assessed and compared over the historical period, then combined to optimise a criterion of interest. Another method consists in applying bias correction to each simulation, once again, by comparing the model's data with historical observations. Model aggregation and bias correction are generally performed by separate research communities, but why not merge the two? Alpha pooling is a statistical bias-correction model based on constructing a cumulative distribution function (CDF) calibrated over the historical period. "The appeal of this method is that with fewer parameters and a simpler procedure, there is less room for error and therefore projections are more robust," explains Soulivanh Thao.

"For several years now, we've been trying to develop links between the variables or grid points to correct several things at once," he adds. "This type of approach would be even more useful for multivariate bias correction." Traditional univariate methods link a single variable to each grid cell on the globe, but this can sometimes fail to maintain consistency between neighbouring points. This original concept combines the benefits of both traditional methods and the results are promising on the reference models.

Popular methods

Statistical models may be major allies in achieving a higher resolution, especially using regression methods - powerful statistical analysis and prediction tools. These methods make it possible to downscale global models by modelling the relationships between large-scale variables and local observations of a variable of interest, while avoiding certain parametrisation biases. Convolutional neural networks are growing in popularity, especially for non-linear cases where traditional methods like decision trees have their limitations.

Excellent for image processing, recognition and classification, and able to process vast amounts of data, neural networks are already widely used outside the scientific community, for example for plant identification applications. "They are well suited to model grids, process large regions in one go and have more accurate results," explains the LSCE researcher. Since these relationships are learned from the historical period, the question remains as to whether they can be transferred to future projections. "In our study, decision forests seem to be the most reliable in terms of projections," says Soulivanh Thao.

"In the bias correction, we generally correct physical variables in relation to all values observed. When this involves correcting a physical object or a very specific meteorological event, it's more complicated," he explains. Cyclones can be particularly challenging to model due to their natural variability, their rapid development and processes which are still not properly understood. To address these challenges, his colleague Davide Faranda developed an original approach by defining dynamic metrics related to cyclone persistence and local dimension. The concept is to sort the cyclones observed in the real world and in the models into different categories and, based on these two metrics, to correct the pressure fields in the model using more traditional bias correction methods. "If we want more realistic cyclone representations in the simulations, we're going to need this kind of method, but it's still very new and requires a lot of development," summarises Soulivanh Thao.

A closer look at rainfall

Carte des précipitations de la tempête Eowyn 20250124

In general, rainfall modelling still involves a great deal of uncertainty. "In recent years, there have been discussions about developing tools to establish correspondence between different precipitation events. The main aim is often to extract common information," says Aymeric Chazottes, a researcher at the Atmospheric Space Observations Laboratory (Latmos – Univ. Paris-Saclay/CNRS/UVSQ/Sorbonne Univ.). He refers to an approach being explored with a view to improving detection and predictions across the Paris basin. This compares rain series with each other, using an improved version of the Dynamic Time Warping (DTW) algorithm. The iterative and multi-scale nature of this version of the DTW gradually refines resolution while avoiding the problem of locking onto details that are irrelevant at the global scale for improving weather forecasts.

Another approach tackles the representation at raindrop level. Rain is governed by complex processes, some of which are studied in laboratories by specialists in fluid mechanics. It is still a random event that is difficult to measure and some of its phenomena remain poorly understood. The Latmos researcher was involved in the study of a new method able to estimate vertical raindrop size distribution simultaneously with vertical wind profiles during light and stratiform rain events, using data assimilation. The approach consists in coupling ground disdrometer measurements, which record raindrop fluxes, with spectral observations from a vertically pointing Doppler radar.

This is done using a data assimilation technique called 4D-VAR (Four-Dimensional Variational Data Assimilation). Commonly used to improve weather forecasts, this method integrates spatial and temporal observations together in a numerical model in order to determine the most likely state of the atmosphere. According to Aymeric Chazottes, "the method worked quite well, but it relies on simplifying many complex issues and ignores some important parameters". Numerical models are therefore emerging as a vital tool for improving our understanding and explaining the physics governing precipitation.

The contribution of deep learning

France has an exceptional radar network and highly reliable weather forecasts. "There is huge potential when it comes to nowcasting," explains Aymeric Chazottes. This refers to very short-term forecasting, for example, anticipating the need to evacuate areas in the event of heavy rain. The priority is then fast models that can deliver almost immediate projects, at the expense of precision. Neural networks are perfectly well suited to this task, as they can learn from massive volumes of data and provide estimates in a short space of time.

These models also comprise many parameters and meta-parameters. The researcher explains that "neural models are not necessarily produced by specialists in meteorology. It is vital to foster interactions between disciplines and to help machine learning experts integrate the orders of magnitude specific to meteorology, so as to optimise the parameterisation and training of models. It is equally important to support meteorologists in choosing the numerical tools that are best suited to their questions and projects."

Furthermore, radar networks are costly to deploy and maintain, so to tackle global challenges, other types of approaches need to be explored. This is illustrated by an opportunistic project on the quantitative assessment of precipitation, using geostationary satellites combined with a retrieval algorithm.

A lever for applied research

Données satellites pour prédiction impact

The data provided by satellites and machine learning models are valuable resources in predicting the impacts of extreme climate events on agriculture and identifying climate change adaptation strategies. David Makowski is a researcher at the Applied Mathematics and Computer Science Laboratory - Paris-Saclay (MIA-PS - Univ. Paris-Saclay/French National Research Institute for Agriculture, Food and Environment (INRAE)/AgroParisTech). He worked on a project that aims to exploit satellite data using deep neural networks to predict maize harvests in the United States and several African countries. "Some regions of the world, like Africa, depend heavily on this cereal and these tools could be a game-changer," he explains. This type of forecasting could potentially have a significant impact, especially since the satellite data used in this project are freely available.

The possible applications are numerous, such as predicting the price of agricultural raw materials, and they are proving particularly effective in optimising climate change adaptation strategies. The MIA Paris-Saclay researcher recalls that "in systems that involve non-linear effects and multiple interactions, simple statistical methods, such as linear regression, quickly reach their limits." In contrast, machine learning techniques can be used on ambitious projects combining multiple objectives, like optimising the allocation of soybean crops in Europe. A complex problem that reflects a real challenge to reconcile high production objectives and resilience to climatic variations, while reducing European dependence on soybean produced in South America, currently the leading supplier of soybean for animal feed in Europe.

In this type of complex issue, machine learning provides a decisive benefit because it does not require researchers to formulate a specific equation or anticipate all possible interactions. Machine learning uses a wide range of algorithms and configurations, retaining the agility of non-parametric approaches while offering good predictive capabilities. Valuable flexibility in this kind of context.

Endless application potential

The last decade has been marked by the growth in new algorithms and unprecedented computer power. This combination has brought new opportunities for scientists, a much more enhanced "toolbox", capable of producing more reliable predictions that can be applied to various fields, often related to very real issues. According to David Makowski, "one of the major challenges in the coming years is to identify areas of vulnerability to climate change - hotspots - so we can be better prepared to adapt." Numerical tools can assess multiple forms of vulnerability, whether these relate to infrastructures, ecosystems or populations, to inform the planning of adaption measures.

Modèles de prédictions pour l'agriculture

"In agriculture, the potential is enormous," adds David Makowski. He gives the example of rice, which is a staple food for humans, but poorly adapted to excessively high temperatures. Using the "random forest"-type machine learning algorithm, the researcher quantified the effects of climate change (drought, high temperatures) on rice quality and pinpointed adaptation strategies. He stresses that "whatever the project, the challenge is understanding whether the predictions are accurate or not. To choose the best algorithm, different predictive algorithms must be tested using robust data." He believes that machine learning satisfies a need for the scientific community, but that this requires high computing power which sometimes has a significant energy cost.

Having become essential, numerical methods are also paving the way for new fields, such as attribution science: a field of study with particular societal value, that investigates to what extent extreme events can be linked to climate change. "And this technical trend doesn't just apply to the geosciences, the use of numerical methods in economy and sociology considerably disrupts the methods of researchers in social sciences. It's in these disciplines that we'll see major impacts in the coming years," concludes David Makowski. It would be difficult to imagine the world of research without these tools. They are helping to improve our overall understanding of the Earth system and its future development, design and optimise climate change adaptation strategies and even anticipate risks and strengthen the resilience of socio-ecological systems in the face of an uncertain future.
 

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