DEIXIS 2007-2008 THE DOE CSGF ANNUAL
Lawrence Livermore National Laboratory
By Victor D. Chase
“Earthquakes don’t kill people,” seismologist Arthur Rodgers says. “Buildings do.”
“There are casualties in earthquakes because buildings collapse, freeway sections collapse, and bridges go out,” says Rodgers, a member of an earthquake modeling team at the Department of Energy’s Lawrence Livermore National Laboratory (LLNL). In essence, “We are vulnerable to earthquake damage because we choose to build and live near places where earthquakes occur.”
To help understand and prevent such devastation, a team of computational scientists, applied mathematicians and seismologists at LLNL has created an earthquake simulation model as part of a larger Serpentine Wave Propagation (SWP) project. The project, headed by applied mathematician Anders Petersson, looks at the propagation of waves in nature. Whether they’re seismic, electromagnetic, or sound waves, they’re all governed by essentially the same mathematical equations.
The team spent several years developing the advanced mathematics and algorithms necessary to run computer models of wave propagation. For its first practical application, the team used the software to model what happened during the most famous and damaging earthquake in U.S. history.
It began at 5:12 a.m. on April 18, 1906. San Francisco Bay area residents were awakened when the San Andreas fault, a 296-mile fissure beneath the Pacific Ocean a few miles along the California coast and off shore, slipped. The displacement of a few meters along that fault line, where the Pacific and North American tectonic plates meet, was enough to set off one of the most monumental quakes in recorded history. After a 20-second foreshock, the full power of the quake was felt for about one minute. It would have measured 7.9 on the Richter scale — if the scale had existed then.
The shock was felt from Coos Bay, Oregon, to Los Angeles, and as far east as central Nevada. In all, the area of devastation was about 400 miles long, and 30 miles on either side of the fault zone. The quake and the resulting four-day San Francisco fire killed about 3,000 people, left 225,000 homeless, and destroyed about 28,000 buildings.
Poorly Understood
Earthquakes were poorly understood and little studied before the San Francisco disaster. The 1906 event put an end to that and marked the beginning of the science of seismology in the U.S. and gave rise to a more quantitative approach, applying physics and mathematics to the problem. Shortly after the quake, damage throughout the region was studied and quantified on the Modified Mercalli Intensity Scale. Unlike the Richter scale, the Mercalli scale does not require instrumentation. It rates a witness’s impressions and physical damage to structures. Scientists can use Mercalli scale information to backtrack and determine the kind of ground velocities corresponding to the reported destruction.
The report has proven invaluable to later earthquake investigators, including the SWP team, which developed its computer simulation quake to mark the quake’s 100th anniversary. Conducted under the leadership of the U.S. Geological Survey (USGS), the research also involved scientists from Stanford University, the University of California at Berkeley, and URS Corp., a worldwide engineering firm. DOE’s Office of Advanced Scientific Computing Research supported the SWP group’s modeling work.
Each participating group created its own model of the quake using its own methods. The results were quite consistent, proving the exercise’s value, Rodgers says.
The exercise took almost two years to complete because of the complexity involved in creating a computer simulation of an earthquake. The rarity of major earthquakes in the 7.0 to 7.9 magnitude range, means there is less empirical data about them.
The findings of the centennial study were presented at the 2006 meeting of the Seismological Society of America, which was held in San Francisco to commemorate the100th anniversary of the famous quake — and of the society’s founding.
To develop their simulations, each of the participating groups began with a USGS-created geological model of the Greater San Francisco Bay area. The model characterized rock and soil properties and was developed from years of study of seismic data, drilling, and tomography up to a depth of 50 kilometers. The data was crucial to creating a computerized picture of the earthquake in progress because different types of earth have dramatically different responses to the spreading shockwaves.
To create its model, the SWP team used two LLNL-based supercomputers:
Doing the Math
One reason creating an earthquake model is difficult is that “a continuous elastic body, such as the earth, has an infinite number of degrees of freedom corresponding to motion at each point in space,” Petersson, the SWP team leader, says. “So before the motion of a system with an infinite number of degrees of freedom can be calculated in a computer, we need to reduce the number of degrees of freedom to a finite, large number through a process called discretization.”
The group used a finite difference discretization method. It broke the area being modeled, known as the computational domain, into a series of equally-spaced grid points.
“In our calculations we used a grid spacing of 125 meters, which corresponds to something like 2.3 billion grid points,” Petersson says. “At each of these grid points you have three degrees of freedom, so you get about 6.8 billion degrees of freedom. You reduce your infinite number of the degrees of freedom to 6.8 billion,” which “is still fairly large.”
The earthquake’s motion also is integrated in time, which also must be discretized into a manageable number. The group took equal steps in time to simulate the first 300 seconds of earthquake motion. “Each of these steps is 0.01 seconds, so you take about 30,000 time steps in the calculation,” Petersson says.
In discretization, “All the derivatives in the partial differential equations are replaced by what are called divided differences, and that converts the original mathematical equation to a set of algebraic equations, and those can be solved in the computer,” Petersson says.
Without discretization, a computer cannot solve the original equations because they are too complicated — even for a supercomputer. “The computer can only deal with simple operations like addition, subtraction, multiplication and division. The partial differential equation involves very complicated relations between how the solution varies in space and time,” says Petersson.
The Freedom is in the Details
But there is another factor to complicate the discretization process: stability. “If the method is unstable then perturbations due to round-off errors in the computer will accumulate during simulations, and they can make the computed result completely useless,” Petersson says.
That’s where the mathematician’s expertise comes in. The key is to develop a mathematical theory that guarantees the stability of the process “so you know before you start your calculation that it is not going to go unstable, or ‘blow up,’ as we also say,” Petersson adds. To do so, “Mathematically, you study how perturbations propagate through such a calculation without actually computing the solution. You can then estimate how large these perturbations can become in a calculation.”
This is accomplished the old-fashioned way. “With pen and paper you can analyze the properties of your numerical method. And there are various ways you can modify your finite difference method so there’s not just one prescription, there’s a lot of freedom in how you do all the details. So you’ve got to figure out the way to deal with all the details such that you can guarantee that it is stable. And that,” Petersson says, “is the challenge.”
Earthquake in a Box
Rodgers describes the SWP modeling effort from a seismologist’s perspective. “Think of it as though we built a box that represents the Bay area in three dimensions,” he says, “and we put an earthquake in the box, and the earthquake sets the box in motion. By doing so we were able to put a simulated seismic station that measures the ground motions as a function of time at any place, which allowed us to compute the ground motion anywhere.”
Modeling the San Francisco earthquake has significance beyond commemorating the event. The ability to create a computer simulation of such a complicated occurrence enables scientists to model other earthquakes before they happen. Architects and civil engineers can use data gathered from those models to design structures that withstand tremors.
What DOE’s supercomputers and scientists can’t do is predict exactly where or when earthquakes will strike.
“We know that earthquakes are going to happen,” Rodgers says. “The problem is that we have only been looking at earthquakes in detail for about 100 years. The return times of large earthquakes are hundreds, if not thousands of years, so we haven’t got a statistical sample to allow us to do meaningful statistics.”
Nonetheless, “In the Bay area the most likely next earthquake will be along the Hayward fault,” he says. This supposition is based on geological studies indicating there have been 11 earthquakes along the fault at intervals averaging 140 years. The last such quake occurred in 1868, making 2008 the 140th year.
Using modeling, the researchers “can put in a hypothetical Hayward fault magnitude 7 earthquake, and see what happens,” Rodgers says. Although models cannot tell experts precisely where along the fault the earthquake will start or in what direction it will run, “We can do lots of simulations to look at how the ground motion might vary depending on those types of factors.”
Such scenarios are valuable because “Our data set of actual large earthquake shaking is limited,” Rodgers say. “So this modeling effort is very important because it allows us to, in the safety of our computer, compute the shaking that would occur if an earthquake were to happen on a specific fault of a certain size within a certain geology.”
From the Soil Up
Such projections are especially needed to avoid accidents at the many nuclear power plants being considered to meet increasing worldwide energy demands. Seismic safety of nuclear power plants is guided by observed, as well as computed, ground motions. The same computer modeling can also be used to simulate potential damage should an earthquake impact nuclear storage facilities, such as the controversial Yucca Mountain site. “Then you would know how to design containers to withstand the possible motions of the Earth,” says Petersson.
Now that the SWP team has created a program that models how earthquake waves propagate from the source through rocks and soil to the foundations of buildings, the next logical step will be to follow those waves up from the soil through complicated structures, such as nuclear power plants, airports, and bridges, to learn how they will respond to the shaking of an earthquake.
“The foundations of buildings are embedded within soil so they need to be modeled together,” said Rodgers. “We would like to be able to model what’s called the soil/structure interactions.”
Meanwhile, the SWP team recently made some waves of its own by receiving an internal award from the DOE’s Energy and Environment Directorate for its work on the 1906 earthquake.
Anders Petersson is an applied mathematician in the Applied Mathematics group in the Center for Applied Scientific Computing (CASC). His research interests lie in the areas of grid generation and numerical solution of partial differential equations. Dr. Petersson earned his doctoral degree in Numerical Analysis from the Royal Institute of Technology in 1991. He joined the Lawrence Livermore National Laboratory in 1999.
Arthur J. Rodgers is a physicist and Group Leader of the Seismology Group in the Atmospheric, Earth and Energy Department at Lawrence Livermore National Laboratory. He received his Ph.D. and M.S. in Physics from the University of Colorado. Dr. Rodgers’ research interests include computational seismology, earthquake ground motion simulation and nuclear explosion monitoring. He joined Lawrence Livermore National Laboratory in 1997.
Contact:
Anders Petersson
petersson1@llnl.gov
Arthur Rodgers
rodgers7@llnl.gov
Practicum Coordinator:
Tom Epperly
tepperly@llnl.gov
The Krell Institute
http://www.krellinst.org/