Understanding what lies beneath the Earth’s surface is crucial for fields like archaeology, environmental studies, and civil engineering. However, the tools we use to explore underground, such as electrical imaging techniques that scan in two dimensions, can be seriously affected by uneven ground. When working in hilly or rocky areas, the measurements are often distorted by changes in elevation, inconsistent spacing between sensors, and variations in the electrical current used. These factors can create unwanted noise in the data, making it harder to understand what is really underground.

To address this challenge, Dr. Andrés Tejero-Andrade, Dr. Aide López-González, Professor José Tejero-Andrade, Dr. René Chávez-Segura, and Professor Denisse Argote from Universidad Nacional Autónoma de México, Tecnológico Nacional de México, and Instituto Nacional de Antropología e Historia developed a new method to clean up this noisy data. Their improved method helps make data collected in difficult landscapes clearer and more accurate. The study was published in the journal Mathematics.

The technique involves three main steps. First, they adjusted the recorded voltage readings to account for variations in current strength. Second, they corrected the measurements for differences in how far apart the sensors were placed. Finally, they applied a special kind of mathematical filter to smooth out random spikes and dips in the data. This filter uses a type of formula known as Legendre polynomials, a sequence of mathematical expressions that help in fitting curves to data, and it works by running through the data multiple times to remove unwanted variations. When tested at an archaeological site in Mitla, Oaxaca, and in a polluted area north of Mexico City, the method produced much clearer and more accurate images of what lies underground.

The results are impressive. At the Mitla site, older methods failed to detect what might be a buried chamber under a historic chapel. Using the new method, a rectangular underground structure became visible in the electrical scan. In the polluted zone, the updated technique showed clearer signs of soil contamination, which matched with areas known to have high levels of benzene in the ground. In every case, the improved process reduced the amount of confusing or unusable data and made the images more understandable.

“The least-squares filter with Legendre polynomials, which minimizes the average of the squares of the errors to smooth data, effectively removes random noise while preserving the shape and peak of the electrical signals,” said Dr. Tejero-Andrade. “Even when used without additional correction, it significantly improves image definition.” Dr. Tejero-Andrade added, “But when voltage normalization, which ensures all readings are adjusted to the same baseline, and geometric factor correction, which accounts for differences in spacing and setup, are also applied, we see a complete alignment of modeled anomalies with actual subsurface features.”

This new approach could transform how scientists and engineers analyze data collected in rough terrain. By dealing with the sources of error more thoroughly, the team created a process that keeps the important signals while removing the confusing ones. Even compared to other well-known filters like the moving average, which smooths data by averaging nearby values, or the Savitzky-Golay method, which fits segments of the data to a polynomial to preserve detail, their approach produced cleaner results with fewer mistakes and better alignment with known underground features.

The key takeaway is simple but important: the better the input data, the better the final picture. By using this smarter way to prepare and correct their data before analysis, researchers can make more confident decisions—whether they are locating ancient ruins or mapping out pollution in the soil. The study shows that applying thoughtful corrections and careful filtering can provide a much clearer window into the ground below.

Journal Reference

Tejero-Andrade A., López-González A.E., Tejero-Andrade J.M., Chávez-Segura R.E., Argote D.L. “Smoothing Filter and Correction Factor for Two-Dimensional Electrical Resistivity Tomography and Time Domain-Induced Polarization Data Collected in Difficult Terrains to Improve Inversion Models.” Mathematics, 2025; 13(5): 866. DOI: https://doi.org/10.3390/math13050866

About the Authors

Andrés Tejero-Andrade: Graduated with a degree in Geophysical Engineering from the National Autonomous University of Mexico (UNAM). He earned a Master of Science degree from the University of Toronto, Canada, and a PhD in Geophysical Exploration from UNAM.

Aidé E. López-González: Studied Geophysical Engineering at UNAM. He earned a Master’s and Doctorate in Geophysical Exploration from the Institute of Geophysics of UNAM.

Rene E. Chávez-Segura: Graduated with a degree in Physics from UNAM. He is pursuing a Master’s degree in Science and a PhD in Geophysics from the University of Toronto, Canada.

Dennise Argote-Espino: Graduated from the National School of Anthropology and History. She earned her Master’s and PhD degrees in Geophysical Exploration from the Institute of Geophysics at UNAM.

José M. Tejero-Andrade: Graduated with a degree in Physics from the National Polytechnic Institute (IPN). He earned a Master of Science degree from IPN. He earned a PhD from the Institute of Chemical Kinetics and Heterogeneous Catalysis of the National Research Council  (CNR), Lyon, France