2 April 2013

## Simple Math Gives Readers X-Ray Vision

*Posted by kramsayer*

**By Mohi Kumar**

As a staff editor for *Eos*, I see all types of articles pass my desk, from those littered with the alphabet soup of undefined acronyms and the jargon best reserved for textbooks, to lovely pieces that describe the science of atmospheric rivers and the emerging field of isoscaping.

But a few weeks ago, a gem came across my desk. I barely needed to touch it, and after reading it I experienced a stillness of the mind. You know those magician’s black boxes we build in our heads, where complicated stuff goes in, hands are waved, and—poof!—useable information comes out? I knew that one of those floating in my mind was just rendered transparent.

The article I am referring to is “Shallow Versus Deep Uncertainties in Natural Hazard Assessments,” by Seth Stein, a seismologist at Northwestern University, and his late father Jerome Stein, a former professor at Brown University. The article was published today (Tuesday, 2 April) in *Eos*— the newspaper of the Earth and space sciences, published by the American Geophysical Union.

Don’t let the staid title fool you. The piece takes the complex issue of how scientists characterize the chance that a natural hazard will occur and boils it down into fundamentals that are understandable to anyone who can envision a container with two different types of marbles in it.

“Imagine an urn containing balls…in which e balls are labeled “E” for event and n balls are labeled “N” for no event,” the authors write. “The probability of an event is that of drawing an E ball, which is the ratio of the number of E balls to the total number of balls.”

Pretty basic so far, right?

The Steins continue: “Two models describe how the probability of an event changes with time. One is sampling with replacement—after drawing a ball, it is replaced. In successive draws, the probability of an event is constant or time independent…Events are independent because one happening does not change the probability of another happening. Thus, an event is never ‘“overdue’ just because one has not happened recently, and the fact that one happened recently does not make another less likely.”

But in their second model, they change the ratio of E balls to the total number of balls by adding extra E balls when no event occurs and removing them when one does. Now “events are not independent because one happening changes the probability of another.”

If you are charged with selecting a ball every year, what emerges as a consequence of your selections mirrors the behavior of natural hazards over time. And the scant math involved is certainly far less complicated than figuring the odds of your team winning a sporting event or the chance that the next card dealt will help your poker hand.

The issue then becomes one of figuring out how many extra balls to add or remove. These quantities are difficult to quantify for earthquakes and volcanoes, rendering their uncertainties to be large, or “deep” as the authors put it. They go on to characterize sources of deep uncertainty

Other than lucid writing and their article’s opening line, which the authors borrowed from Shakespeare, what makes the piece so successful is its characterization of natural hazards through a classic scenario used in probabilistic studies of drawing balls from an urn. This is more than an analogy. As they note, it is a model.

But in a world where models are becoming increasingly complex, where calculations involve supercomputers and terabytes of data, the Steins’ model gives you x-ray vision into the black box of hazard assessments.

What other insights into complex science can be illuminated through simple math? Let us know your ideas!

**–Mohi Kumar, Eos Science Writer/Editor**