Broadening the Family Tree

River network study links mathematics, statistics and science

11/30/2009 - By: John Trent
Zaliapin Photo by Jean Dixon

On the face of it, the connection between the work of a mathematics and statistics professor and the world of river ecology would seem tenuous at best.

And yet, according to Ilia Zaliapin, an associate professor of mathematics and statistics at the University of Nevada, Reno, there are more connections than meet the eye.

So many that the merging of the principles of applied statistics with the natural world come together as easily as branches growing on a tree.

Trees, in fact, hold one of the keys to what Zaliapin is studying.

Zaliapin, as part of a major grant from the National Science Foundation totaling in excess of $600,000 and involving several institutions, is bringing these two seemingly disparate elements together in a compelling way.

He’s hopeful his research will help to explain such important questions as environmental transport along the world’s river systems and flood occurrence. This new direction in earth-surface modeling could hold the key in better predictive understanding of river systems’ precipitation, sediment loads, nutrient and pollutant content and health of associated food webs.

“With this project, we’re trying to relate mathematical and applied statistics principles to hydrological principles,” Zaliapin explains. “There was a huge amount of research that started in the 1940s and 1950s on the geometry of rivers, but we still lack theoretical and modeling frameworks to connect geometry with processes such as how snow travels from mountains to rivers.

“We’re focusing on understanding an area of science that involves transport along river networks by applying principles of applied statistics and general theory of complex networks.”

Zaliapin’s $224,075 grant is just one part of a four-team, four-institution collaborative effort including researchers from the University, the University of Minnesota (Principal Investigator: Professor Efi Foufoula-Georgiu), UC-Berkeley (P.I.: Professor William Dietrich) and UCLA (P.I.: Professor Michael Ghil) that is studying Envirodynamics on River Networks.

For a mathematics professor such as Zaliapin, the marriage of numbers and river networks makes for a natural test case. River networks, he says, are examples of fractals, or fragmented geometric shapes whose parts are smaller versions of a whole. Fractals occur everywhere in nature, Zaliapin says.

“The branching structure of trees, for example, are fractals,” he says. “Rivers have a clear tree-like structure, which lends itself so well to what we are doing.”

As Zaliapin notes, river networks in many ways are the perfect laboratory for a mathematician to develop numerical models that help explain many of the networks’ physical and geometric processes.

River networks provide fragmented space for all things which move through them, allowing researchers like Zaliapin to construct theoretical representations with clear mathematical rules rooted in the theory of self-similar trees and nearest-neighbor aggregation.  These tools can, in turn, help hydrologists make key projections for the health of wildlife, or how the food web can be impacted by disruptions, either man-made or naturally caused, in the flow of a river.

The work, Zaliapin explained, has clear implications for other fields, such as disaster preparedness.

“It can be applied to many different processes, not just rivers,” he says. “It’s most obvious for rivers because they have such a tree-like structure. But the same principles can be applied to almost any field. Using this tool, we can solve many problems that may not be solved by conventional means. We can talk about blood systems or lung systems or lightning or valleys on Mars or even gas dynamics – they all have, surprisingly enough, the same, similar, branching structure.”

It is this wider application that led Zaliapin to find collaborators on the University campus involved with seismology. Both John Anderson, the former director of the Nevada Seismological Laboratory, and Corne Kreemer, of the Nevada Bureau of Mines and Geology, have been working with Zaliapin on another project that is looking at statistical modeling of seismic energy release. Their work has implications for seismic hazard assessment.

“Everyone realizes that major floods and big earthquakes are some of the most damaging natural disasters that occur,” he says.  “Terrible damage can occur from a major, extreme event. That is why it’s not only very important to develop the scientific part of this project … we need to deliver information to the government and the public. The collaboration on our campus has been great.

“I’ve been warmly greeted by the Seismological Lab and the Nevada Bureau of Mines and Geology.  Together with my home department, they really are supportive of what our studies are hoping to accomplish.”

Zaliapin’s studies also heavily involve his students. During the completion of a previous grant on a dynamics of seismicity, Zaliapin was able to support five students who played prominent roles in the research, including making nine presentations throughout the country. One Honors Thesis was also completed thanks to the grant.

“Thanks to funding like this, we are able to have our students contribute in meaningful ways,” he says. “It’s much better than just explaining it in the classroom. We hope this is just the beginning of developing many more new methods and techniques.”


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