Estimate the flow of a river on a planetary scale for the first time

Fresh water is vital for life, for our society and cannot be replaced. Economic development, agriculture, industry, and a growing population are creating tensions around these limited resources, even more so because of climate change.

80% of the world’s freshwater (90% in some countries), 19% of industry and only 12% of domestic use (most of which are related to household use such as washing-linen, dishwasher, watering, bathroom and to a lesser extent related to drinking). To date, 1 in 2 inhabitants of our planet has lived in areas affected by severe water shortages for at least one month a year. France is also experiencing tensions over the use of water on its own scale. After the dry and hot winter of 2022 and then the spring, the prefecture of the Alps-Maritimes, for example, had to keep two-thirds of its municipality (including some large cities) on “drought alert” for several months, from late winter.

As seen from the natural risk angle (which is three times more frequent overall in 2020 than in 1975), “excess water” represents a significant hazard. Floods represent about half the risk for 2021 alone, with more than 50 major events and more than 80 80 billion in losses. These results are getting worse and will continue to get worse with climate change.

The water cycle is still not well understood and flows in small amounts, including surface currents (currents) flowing towards the sea through water currents (rivers). Rivers and streams act as veins in our terrain.

Human activities depend on the amount of water available in abundance and also change resources. In general, the policy of managing a dam for agricultural development in a region can create further deficits in a region. Tensions can then arise between regions or users (agriculture, industry, population). Examples include conflicts between Israel and the Arab states that are exacerbated by water shortages (such as disputes around Lebanon’s Litani or the small Yarmouk River in the Golan Heights), tensions between countries bordering the Nile, or Mexico, and tensions between them. The United States in Rio Grande and around Colorado.

Estimating the flow of rivers on our planet is a big challenge both from a scientific point of view and from a socio-economic point of view. Contrary to what we might think at first glance, the currents are far from well-estimated on the scale of the planet and the task is not entirely clear.

Mathematical measurements and models

The main variable for measuring surface water flow is river flow Q (m)3/ s), Q = AU, A (m2) Being section across the river, U (m / s) is the average speed in this section.

Image of river flow calculation.
Jerome Monier, Contributed by the author

Flow measurements are available daily or even hourly in industrialized or densely populated areas of the world, for example in France via the Visigruce network. Conversely, in LDCs, data does not exist; The flow is therefore assumed to be very bad.

It is possible to estimate river flow through mathematical models and numerical calculations. To do this, on the other hand, it is necessary to know the depth of the river (in h), the shape and nature of its bed, and the topography of the surrounding terrain. Without field measurements, the depth of a river remains unknown (what is the depth below?). In addition, to be able to use these numerical models requires physical parameters such as the friction coefficient of the flow in the ground.

One challenge then is how to make mathematical estimates, numerically h river depth, its physical parametrization and finally its flow Q, from the available measurements which are usually only the height of the water surface (and not its depth) at certain points in the river.

Measure the water level from space

To make up for the lack of measurements in our field, which in most parts of the world, spatial observation of rivers should soon be a solution.

The SWOT satellite, which will be launched in the autumn of 2022, will for the first time be able to measure river water surface elevation, more than 100 meters wide and for more than 90% of the world’s rivers, extending 213,500. About ten kilometers. The frequency of the measurement will be about ten days (depending on the latitude of the river). The spatial density of the measuring points will be about 250 m.

From this measurement of water surface height H (m), the scientific challenge is to convert these measurements into flow values ​​Q (m).3/ s), knowing that in non-instrumental areas, flow speed and river depth are unknown! ..

INSA – Institut de Mathématiques de Toulouse, INRAe, University of Strasbourg – ICUBE and CS Group (CNES funding) conducted multidisciplinary research in applied mathematics, computational science, hydraulics and hydrology. Their flow from measurement. This challenge is being addressed based on mathematical models of fluid mechanics (e.g., the equation of Saint-Venant XIX).e Centuries) has been revisited in this particular multi-scale and observational context, the kind of mathematical method of control best used to control the motion of a robot or to determine the initial state of the atmosphere before the weather forecast and deep learning (“artificial intelligence”).

These scientific advances are applied to obtain calculation algorithms. Our algorithm called HiVDI is available in our calculation software for hierarchical variable discharge interface, which is of course technical, but open to all (DassFlow research software).

The present estimates are based on purely numerical measurements from a CNES-NASA simulator of future SWOT instruments and three comparative algorithms (including two US) with different methods.

The results of the calculations make it possible to expect approximate real-time estimates of non-instrumental river depths and, above all, relatively accurate estimates of flow (within about 30%). Such estimates should be available after a full year of time for satellite overflight, model calibration and learning.

Based on these estimates of river flow on the planet’s scale, will we be able to refine our knowledge about the water cycle? Can we better estimate the impact of different uses of specific waterways, manage them better? These questions are part of the challenges of this research in applied mathematics and computational science.

This article is part of the “Great Story of Science in Open Access” series published with the support of the Ministry of Higher Education, Research and Innovation. To find out more, visit

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