June 13, 2021

# Mathematics that improve European satellite navigation systems | Science

Mathematics is essential to equip companies with competitive advantages. It is something that some companies (mainly those based on technology) have known for decades, but now for the first time there is a report that quantifies it 10% of GDP and 6% of employment in Spain are due to mathematics, according to the study carried out by Afi (International Financial Analysts) commissioned by the Strategic Network in Mathematics (REM). They are striking data but, in practice, how do companies use mathematics?

Depending on the type of business, the influence of mathematics is greater or lesser. In the field of satellite navigation systems, they are fundamental. This technology allows to calculate the position of any point on the surface of the Earth from the distances measured to three satellites of the network, of which the position is known. Since the distance to the satellites is calculated from the time it takes for the signal to travel from the satellite to the receiver, it is necessary to have information from the satellite clock. These calculations are executed in a ground processing system, which makes very accurate estimates of the position of the satellites and their clocks, using mathematics.

In Europe, the radionavigation and positioning program Galileo aims to provide the Union with a technology independent of US GPS and Russian GLONASS. The Spanish company GMV develops the algorithms which calculate the information transmitted to the end user. Currently, one of the main objectives is to improve the availability of the service to the user (that is, at any point and time can receive navigation information), the accuracy and reliability of the information provided; which is essential for applications such as vehicles without a driver, for example.

To improve the accuracy of the calculated position of the user it is necessary to increase the accuracy of the position and clock of the satellite system. These data are estimated using measurements at reference stations, whose position is known. The more measures that are used, the better the approach, but, at the same time, the longer the processing time. But this information must be updated and provided to users every 10 minutes, so the time to perform the calculation is very limited. Mathematical creativity was key to increase the amount of data to be processed in the time available. In particular, the mathematics allowed to decompose the matrices involved in the estimation process (that when increasing the number of data, they became enormous) so that they were more manageable computationally.

GMV also collaborates in the European system EGNOS, which provides corrections to improve the information of satellite positions and watches of the GPS system. In this case, the purpose was to increase the availability of service in cases of ionospheric storms. The ionosphere is a region of the Earth's atmosphere in which there are a large number of free electrons that cause delays and disturbances in the electromagnetic signals that pass through them, such as those traveling from the Galileo and GPS satellites to the users' receivers.

EGNOS makes an estimate of the delay introduced into the signal by the effect of the ionosphere and also an estimate of the reliability of these corrections, which is known as integrity; but when there are ionospheric storms, the estimation becomes complicated, even forcing to declare the EGNOS service as not available. Employing numerous mathematical and statistical concepts, such as Kalman filters, the management of covariances, the calculation of probabilities, etc. it was possible to obtain a good estimate of the maximum error of the corrections provided to the user, and thus reduce the uncertainty of the approaches and continue offering the service.

These two programs show that to overcome the known limits in science and engineering we need more powerful and reliable machines and instruments that allow us to collect more data and with better quality, but also improvements in the algorithms that process this data, in order to extract also more and better information. Big data, machine learning or process automation are essential to achieve competitive advantages. In addition, to be done properly and effectively, we need professionals who understand the mathematical foundations that support these technologies.

Esther Sardón Pérez is the head of the Galileo Operational System division of GMV

Coffee and Theorems is a section dedicated to mathematics and the environment in which they are created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share points of contact between the mathematics and other social and cultural expressions and remind those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: "A mathematician is a machine that transforms coffee into theorems".

Editing and coordination:Agate Rudder (ICMAT).