# Update

## Math Provides Better Way to Measure Subsurface

The one thing you can count on in life is that math is absolute. Except when it’s not.

Two plus three will always equal five. But when tackling more complex problems, like the amount of contamination in groundwater following a gasoline or dry cleaning fluid spill, computation can get tricky.

That’s because problems of this caliber tend to be nonlinear in hundreds or millions of dimensions and can involve mind-boggling twists and turns between point A and point B. In the end, researchers end up relying on approximations to arrive at an answer.

Research by Jodi Mead, professor of mathematics at Boise State University, and co-principle investigators John Bradford, professor of geophysics at Boise State, and Rosemary Renaut, professor of mathematical and statistical sciences at Arizona State University, is aimed at making fewer approximations and moving closer to a complete solution.

Backed by a three-year \$360,000 grant from the National Science Foundation’s Division of Mathematical Sciences, the team will work on “Collaborative Research Computational Techniques for Nonlinear Joint Inversion.”

That means creating algorithms that allow researchers to measure the Earth’s subsurface (never a precise task) by combining data that rely on different aspects of the physics. Currently they are limited by the tools used to gather measurements: radar, a small-scale method, and electrical resistivity (or conductivity), which works on a larger scale. Questions involving the two ends of the spectrum are solved using different approximations.

“In applied physics, we make approximations of underlying mathematics to make a problem solvable,” Bradford said. “One kind of approximation is used to solve for the low range, and another approximation is for the high range. If we could combine them, we could work it out as one big problem and ultimately arrive at a better solution.”

The question, then, is how to add the necessary measurements while guarding against incomplete or inconsistent information. To answer that, the research team will use nonlinear inverse theory — a mathematical theory describing how to take measurements and estimate a physical model.

“Nonlinear problems are very hard,” Mead said. “Often we know exact solutions to linear problems, but almost never know exact solutions to nonlinear problems.”

Solving this problem will allow for immediate practical use in many areas. For instance, in Idaho, where snow is the primary water storage, it can be used to obtain better measurement of snowpack, as well as improving our understanding of how snow is turned into runoff and how it enters the aquifer.

Bradford said it also could provide better data for projects such as his current investigation in Bénin, where saltwater is intruding into the area’s freshwater supply. More accurately measuring the subsurface would aid researchers in understanding and correcting the problem.

The algorithms and analyses developed through this project will be assessed through imaging of the near subsurface at Boise Hydrological Research Site. Two students, a doctoral and a master’s student, also will be involved in the project.

This material is based upon work supported by the National Science Foundation under Grant Nos. 6PRJ000303 and 6PRJ000304 to Boise State University. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.