Non-linear mathematical techniques could lead to better flood forecasting

Although the world in which we live in is non-linear, or multi-dimensional, engineers and scientists have long used linear mathematical formulas to create models to predict physical phenomena such as the infiltration of water through soils or flooding.

But existing theories based on linear models do not accurately portray what actually occurs in nature, claims Temple University civil and environmental engineering professor Sergio Serrano, Ph.D.

In the September issue of the Journal of Hydrologic Engineering, Serrano outlines new mathematical procedures, or techniques, to produce analytical solutions of the complex, non-linear equations of water flow in soils. These new techniques, says Serrano, will help with the development of more accurate and more efficient flood forecasting and contaminant propagation predictions.

In his study, “Modeling Infiltration with Approximate Solutions to Richard’s Equation,” Serrano says that although a phenomenon such as water flow is non-linear, we try to solve it numerically, which linearizes the solution. “What we do is assume this phenomenon is linear and try to solve it using linear equations,” he says. “For instance, we come up with a model that shows a contaminant plume in either soil or water that is perfectly symmetrical and doesn’t have any of the features that we observe in nature. But if you actually observe a plume in nature, it is not symmetrical and it has a long back-tail that traces back to the source of the contaminant.

“Now, by using these new mathematical methods or techniques, it allows us to consider the true non-linear attributes of this non-linear phenomenon,” Serrano adds. “We now can develop a model that actually describes what is happening in nature.”

Serrano says that linear equations have been used to solve these problems because they are simpler to do. “People think they are using non-linear equations when they use the computer and numerical techniques, but they have not solved the non-linear equation to explain the phenomenon; they have merely numerically linearized the situation,” he says.

Serrano believes that using these new techniques to correctly solve these non-linear equations will help researchers create more accurate models, which will allow scientists and engineers to better remedy environmental problems and better predict flood waves.

“For example, if we assume that the equations that are currently being used to predict flooding are linear, then we will develop a model that predicts a flood downstream to occur at a certain time,” explains Serrano. “In reality, we observe that the flood comes at a much earlier time. So what happened? The flood wave propagates in a true non-linear environment.

“We are beginning to explore the use of these non-linear techniques to understand the phenomenon of water flow and flooding, and we are seeing remarkable differences in what is actually happening in nature as opposed to what was predicted to happen under the current linear methods.”