Nonlinear Models Will Help Reduce Loss of Life and Property from Strong Earthquakes
alifornians won't soon forget the Loma Prieta earthquake that shook San Francisco in 1989, or the wide-spread chaos in Los Angeles after the Northridge earthquake in 1994. Both quakes caused loss of life and extensive damage to highways and buildings, including the campuses at UC Berkeley and UCLA. In both quakes, the ground movement was much more severe and the damage much worse than originally predicted. To understand why, researchers at the Scripps Institution of Oceanography and San Diego State University are developing a more accurate computational model to explain how the ground moves during strong earthquakes.
Jean-Bernard Minster and colleagues are testing a new type of model that describes how seismic vibrations move through the ground. Traditional models are linear; combining two input signals--seismic waves, in this case--leads to an output, the ground shaking, that is a simple function of the two outputs. Doubling an input seismic wave doubles the output ground shaking. In the nonlinear model created by Minster's team this is not true: doubling the amplitude of the incoming waves does not necessarily double the amplitude of the ground motion. The intensity of the earthquakes is therefore much harder to predict.
"If the ground movement from earthquakes is nonlinear, it has important implications for seismic risk mitigation," said Minster, professor at the Institute for Geophysics and Planetary Physics (IGPP) at SIO and leader of the NPACI Earth Systems Science thrust area. "We think we have a method that is easily implemented even in 3-D codes at very minimal extra burden. The method is potentially an order of magnitude--or more--more efficient than classical methods."
The intensity of surface shaking is nonlinear because seismic waves pass through many types of geologic material and many types of underground structures. However, adding nonlinear behavior to models that describe how waves propagate, particularly models in which the waves slow down and lengthen, is a challenging proposition. Most ground motion research uses simple, linear elastic behavior to describe how rocks and soil behave, mainly because nonlinearity is complex and expensive to compute.
Minster, SIO graduate student Heming Xu, and Steven Day, professor of geophysics at San Diego State University, are using NPACI's CRAY T3E at SDSC to drive a nonlinear model of the behavior of rocks and soils. The model has so far been incorporated successfully into one-dimensional and two-dimensional wave propagation codes, and running on workstations, the model has generated results that agree well with experimental observations.
Realistic 3-D models will provide a more accurate picture of how the ground responds during strong earthquakes and allow scientists to better estimate and ultimately better mitigate the effects so as to reduce the loss of life and property in a potentially huge earthquake.
Wave propagation through a nonlinear medium allows energy transfer from one frequency band to another. This figure shows the evolution of the spectrum of a narrow-band 1-D pulse propagating through a nonlinear medium. The peak at frequency 1 decreases in amplitude by losing energy to its higher harmonics. These harmonics, which grow with propagation distance, eventually interact with each other. This pattern of energy transfer is partially masked by energy losses due to the attenuating properties of the medium, resulting in an apparent saturation at the larger distances. In two and three dimensions, the wave propagation is more complex and interpreatation is not always straightforward. (The amplitude of the fundamental is scaled down by a factor of 10 for display purposes).
Following the Northridge quake, in 1996, researchers at UC San Diego, UCLA, UC Riverside, UC Santa Barbara, and Lawrence Livermore National Laboratory joined forces to examine the seismic exposure at several UC campuses under a UC Campus-Laboratory Collaboration program (CLC), complementing the regional studies of seismic hazard being conducted by the Southern California Earthquake Center (SCEC). The ground response simulation by Minster's team is one of several projects sponsored by CLC and SCEC.
Earthquakes can occur at any time, but history has shown that they occur in the same general patterns year after year, and principally in three major earthquake zones; along the rim of the Pacific ocean, where about 81% of the world's earthquakes occur: the belt extending from Java to Sumatra through the Himalayas, the Mediterranean, and out into the Atlantic; and the submerged mid-Atlantic ridge.
At the surface, the intensity of an earthquake at a particular place depends on magnitude, the distance from the earthquake to the epicenter--the point on the surface above the underground origin of the quake--and the local geology at that point. Today, the epicenter, depth-of-focus, and magnitude of an earthquake are recorded by state-of-the-art systems, including seismographs, that precisely amplify and record ground motion.
"With the advent of remote sensing and digital sensor technology, we are collecting data at a truly unprecedented rate," Minster said. "It will take a level of data management capabilities and of computing power that only a partnership like NPACI can provide in order to really make sense of these observations."
Seismologists have observed that earthquake behavior is nonlinear, and that linear models may either underestimate or overestimate the true force experienced at the surface. To create better predictive models of ground activity at the SCEC sites, Minster's team is using data collected from bore holes and ground motion records at the participating campuses, as well as from SCEC databases.
In 1-D and 2-D simulations conducted to date, the team has used the pseudospectral method to reduce dispersion and maintain high accuracy. "This sacrifices computational speed, but we want to avoid ascribing the effects of numerical artifacts to nonlinear processes," Minster said. Standard calculations of energy from an earthquake rely upon an empirical relationship, developed by Beno Gutenberg and Charles Richter, in which the magnitude of surface waves are constrained to a limited bandwidth. Seismologists now know, however, that the energy is concentrated over different bandwidths and at higher frequencies.
With the advent of remote sensing and digital sensor technology, we are collecting data at a truly unprecedented rate. It will take a level of data management capabilities and of computing power that only a partnership like NPACI can provide in order to really make sense of these observations.
Jean-Bernard Minster, Scripps Institution of Oceanography, UC San Diego
To calculate nonlinear transfer of energy from one frequency band to another, the nonlinear model must perform a large number of multi-dimensional Fast Fourier Transforms (FFT) to derive the dynamic fields. This approach, which is rather inefficient for a scalar single CPU machine, can be efficiently implemented on a parallel machine, such as the 256-node CRAY T3E at SDSC.
"The results to date on 2-D and 3-D propagation of compression and shear waves show that most of the nonlinear phenomena observed in one-dimensional workstation calculations remain true at higher dimensions," Xu said. The nonlinear model simulations are forming the basis of Xu's doctoral dissertation. "In addition, some interesting complications arise in higher dimensions, such as mode coupling."
A typical 2-D wave calculation on a 256x256 grid for 2,000 time steps takes eight hours on a Sun SPARC 10 workstation with 10 MB of memory, while a single 3-D time step takes up to five hours. "The memory requirement climbs to two gigabytes for 256x256x256 grids," Xu said. "And meshes of size 512 or larger are needed for realistic simulations that get into the far field characteristics of the wave field." Such a simulation requires about 80 CRAY T3E processors and 20 gigabytes of memory.
The workstation simulations agree well with both laboratory observations on White Mountain granite and Berea Sandstone and with ultrasonic dynamic experiments. " We are in the process of comparing the wave propagation results with published SCEC results of seismic observations," Minster said. "With the availability of NPACI computing resources, we are now ready to proceed with the 3-D modeling."--AF, DH