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\begin{document}

\title{General Earthquake Models}

\author{John B. Rundle\\ Department of Physics\\
Colorado Center for Chaos and Complexity, and CIRES\\
University of Colorado, Boulder, CO 80309 \\
rundle@hopfield.colorado.edu \\
\\
Tom Henyey\\ Department of Earth Sciences and SCEC\\
University of Southern California, Los Angeles, CA 90089-0742 \\
\\
J.-Bernard Minster\\ IGPP, Scripps Institute of Oceanography \\
University of California at San Diego, La Jolla, CA 92093\\
\\
Geoffrey Fox\\ Department of Physics\\
Syracuse University, Syracuse, NY 13244-4100}

\maketitle

\begin{abstract}
General Earthquake Models (GEM) are realistic, high performance
computational 
simulations of earthquake processes on individual and systems of faults
within the earth�s 
crust.  Recently, advances have been made on understanding specific
aspects of earthquake 
processes on various spatial and time scales, however the efforts have
generally been 
carried out by individual scientists, leaving the field of earthquake
science somewhat 
disjointed and not well integrated.  The next logical step will be to
bridge the gaps between 
models of various scales, which will greatly advance our understanding
of earthquake 
processes. 
\end{abstract}

\section*{Introduction}

	General Earthquake Models represent a coordinated effort within the
United States' 
scientific community to develop highly realistic earthquake models for
use in understanding 
the physics of earthquakes.  GEM will develop a set of tools for use on
high performance 
computers to simulate earthquake processes.  By developing these tools
such that the gaps 
between small and large scale models are bridged we will effectively
develop a ``numerical 
laboratory''.  Investigating the physics of earthquakes can then be
approached as an 
``experimental science'' that will complement existing observational and
theoretical 
approaches.  GEM will allow us to test for consistency between models,
to explore the 
effects of friction, segmentation, loading due to plate motion, boundary
conditions, random 
events, and the neglect of sub-grid-scale processes, all of which will
result in more 
physically realistic models, further advancing our understanding of
earthquakes.

	Data will be assimilated to validate the models, and through GEM,
diverse data sets 
can be related to each other by a process known as ``Model-Based
Inference''.  Data 
collection and archiving will not be a part of GEM, however we
anticipate that through 
GEM data needs will be identified.  This will help to drive data
collection efforts in a 
variety of disciplines.  We plan to use  the GEM simulations to develop
new space-time 
pattern recognition and analysis techniques.  These may very likely lead
to the development 
of forecasting and prediction methodologies for the reduction of
earthquake hazard to lives 
and property.

\section*{Spatial and Temporal Scales}

	Processes associated with earthquakes are known to occur over a wide
variety of 
spatial and temporal scales.  Figure 1 illustrates this point.  For
example, frictional 
processes, the primary nonlinearity associated with earthquakes, are
known to be 
physically significant from the molecular scale, on time intervals of
less than 10-8 seconds 
and lengths of Angstroms (�), to plate motion scales, on time intervals
of 106 years and 
lengths in excess of 1000 km.  The physical processes directly
associated with faulting and 
seismology, including nucleation and quasistatic crustal deformation,
occur over time 
intervals of fractions of seconds to thousands of years, and lengths of
meters to a hundreds 
of km.  By contrast, experiments on frictional sliding of rock samples
are carried out over 
laboratory scales extending over typical time intervals of 10-3 seconds
to days, and over 
length scales of cm to m.  Figure 1 indicates that in terms of length
scales, processes of 
faulting, seismology and crustal deformations are as far removed from
laboratory 
experiments on rock samples, as these laboratory rock processes are from
those on the 
molecular scale.    There are no reliable means presently available to
relate results observed 
on one set of scales to those on another.  It is therefore of critical
importance to develop an 
experimental numerical capability that can help to span these large
ranges in scales, so that 
the applicability of physical (and chemical) processes on one scale can
be evaluated for their 
importance on other, distinctly different scales.  An example is the
need to determine 
whether empirical or theoretical friction laws developed to describe the
nonlinearity of 
sliding on one temporal and spatial scale applies to sliding on other
scales.


	Within the region of spatial and temporal scales in figure 1 describing
earthquakes 
(denoted ``faulting and seismology''), there exist additional ``sub-scales''
that describe 
additional heirarchies of processes.  These include, but are not limited
to, the following:

	1.  Individual earthquakes on a single planar fault.  Processes that
may be important 
include inertia; stress transfer via waves and comparison to quasistatic
stress transfer 
calculations; elastic heterogeneity at small scales (elastic wave
scattering); quasistatic and 
dynamic friction; the pore fluid flow; and wearing and evolution of the
sliding surface.  
Important to have friction models for both the quasistatic and dynamic
regimes based upon 
velocity-, slip-, or rate and state-weakening processes.

		Time Scales:		10-1 sec to 102 sec
		Space Scales:		10-1 km to 102 km

	2.  Multiple earthquakes on several faults.  Processes that may be
important include 
quasistatic and dynamic friction; pore fluid flow; dynamic stress
transfer via waves; 
quasistatic/static stress transfer via elasticity and viscoelasticity;
earth rheology; Coulomb 
Failure Functions; Viscoelastic Coulomb Failure Functions; stress
shadows; geometry of 
faults; and how faults influence one another.  

		Time Scales:		10 sec to 108 sec
		Space Scales:		10-1 km to 103 km

	3.  Statistics, correlations, and space-time patterns of earthquakes in
large 
populations of earthquakes on large fault systems.  Processes that may
be important include 
quasistatic friction; pore fluid flow; quasitatic/static stress
transfer; earth rheology; sub-
gridscale faults (sources of noise); plate-tectonic stress loading
processes; and evolution in 
fault geometry due to fracturing of fresh rock.  Since space-time
correlations in populations 
of earthquakes are of primary interest, only processes on time scales
longer than an 
earthquake source time are important, because all smaller time scales
are coarse-grained out 
of the problem. Scaling, critical phenomena and nucleation processes are
clearly relevant on 
these time and space scale, as is obvious from observations of the
Gutenberg-Richter 
relation, Omori's law, and so forth.

		Time Scales:		102 sec to 1012 sec
		Space Scales:		10-1 km to 103 km

	4. Physics of fault gouge formation, role of granula media in geologic
structure of 
faults, heat flow questions.  Processes that may be important include
comminution; 
formation and mechanics of granular gouges; formation of stress
``bridges'' between sides 
of faults; fluid infiltration; fracturing and abrasion of intact rock;
and spatial-temporal 
evolution of fault geometries.  These processes are primarily associated
with the formation 
of geologic fault structures, and their influence on heat flow and
thermomechanics of slip.

		Time Scales:		10-1 sec to 106 years
		Space Scales:		10-6 km to 1 km

	It will clearly not be possible to address all of these problems in a
single proposal, 
or perhaps even within a single computational or physical framework. 
Choices must 
therefore be made as to which problem(s) to address in a given proposal,
and in which 
priority.  These choices and priorities will be determined in
consideration of a given 
funding target of opportunity.

\section*{GEM is an HPC-class Problem}

	The very large range of spatial and temporal scales that these problems
involve 
clearly demonstrate that GEM is an HPC-class problem.  Current evidence
indicates that 
forecasting the damaging earthquakes of magnitude ~ 6 and greater almost
certainly 
depends upon understanding the space-time patterns displayed by smaller
events, e.g., the 
magnitude 3's, 4's and 5's.  At least 40,000 km2 of fault area in
southern California are 
capable of participating in magnitude 6 and greater events.  Hence,
needing a spatial 
resolution of about 100 m to eliminate grid-scale effects and to capture
the physical 
processes of the magnitude 3 events, and a temporal resolution of ~ 100
sec, one arrives at 
the conclusion that as much as 106 grid sites will be necessary for a
maximally realistic 
simulation.  If grid sizes at the 10 m scale are used to capture the
failure physics of the 
magnitude 3 events, then ~ 108 grid sites will be needed.  Using current
computational 
technology, run-time estimates of several months for this problem can be
documented.  

\section*{Scientific Objectives}

	A well-constructed simulation technology should hold the promise of an
iterative 
solution to earthquake problems through a direct comparison of
simulations with a wide 
variety of data. In this context, specific scientific objectives of our
research include:

\subsection*{Objective 1}  Cataloguing and understanding the nature and
configurations of space-time 
patterns of earthquakes and examining whether these are scale-dependent
or scale-
invariant in space and time.  Correlated patterns may indicate whether a
given event is a 
candidate foreshock.

\subsection*{Objective 2}  Understanding the importance of inertia and
seismic waves in determining 
details of space time patterns and slip evolution.

\subsection*{Objective 3}  Understanding the physical justifications for
space-time coarse-graining over 
smaller scales, the role of sub-grid scale processes, and whether these
might be 
parameterized in terms of uncorrelated or correlated noise.

\subsection*{Objective 4}   Developing and testing potential earthquake
forecast algorithms, based 
primarily upon the use of space-time patterns and correlations in the
fault system of 
interest.   
 
\section*{Project Plans}

\subsection*{Models}  The first step in developing General Earthquake
Models will be to identify 
existing codes and models ranging in scale from tenths to tens of
kilometers.  We expect 
that GEM will encompass the dynamics and physics of individual faults,
all the way 
through systems of faults.  This includes fault rupture, the causes of
rupture growth into a 
large earthquake, fault interactions on short and long time scales,  and
the role of rheology, 
heterogenity, random property, and noise.  A centerpiece of early
development will be a 
newly constructed fault-interaction model, to be built by a world-class
team of parallel 
computing physical modelers using modern object/component paradigms. 
The planned 
adaptation of this hugely successful astrophysical N-body algorithm
(``Fast Multipole 
Method'') to the implementation of stress Green's functions on
interacting fault systems 
will represent a major advance, which when completed will provide a
unique capability for 
modeling complex crustal dynamics.  The expected computer requirements
as the model 
matures will exceed 1 teraflop performance, which will become available
through a number 
of computing centers during this time.  Our approach ensures that these
resources are well-
spent, as the N-body algorithm is both accurate and efficient.  In
general, the friction 
models used for GEM will be based upon 1) laboratory experiments, such
as the slip-
weakening or rate-and-state models; 2) computationally simple models
that capture in a 
simple way important aspects of  frictional sliding, such as the classic
Coulomb-Amontons 
law; or 3) statistical mechanics, in which the basic phenomenology of
friction has been 
incorporated on coarse grained space-time scales.  

\subsection*{Pattern Analysis}  Anecdotal evidence accumulated over many
years clearly 
indicates the existence of such space-time patterns in seismicity
data.   The exact nature of 
these patterns, however, has so far eluded identification.  As part of
the project, 
computational tools will be developed to analyze the information content
of both real and 
simulated earthquakes using new ideas about of pattern analysis and
pattern reconstruction 
in complex systems. These include methods based upon log-periodic,
correlation 
dimension, and more recent techniques based upon the discrete
Karhunen-Loeve expansion 
(``Pattern Dynamics'').

\subsection*{Computations}  The current environment is exemplified by
isolated research groups 
that specialize in home-brewed software.  The move to new, more fruitful
collaborations 
marked by shared resources, we will take advantage of the emerging
web-based object 
technology to create an environment where computing platforms, software,
data and 
researchers will easily communicate.  A standard computational
framework, consisting of a 
web-server brokerage system and web-browser based development, data-flow
and 
visualization tools will be agreed to and its components acquired by the
collaborating 
institutions. Because this is a rapidly evolving field, we defer the
choice of tools to the 
initial phase of the work, and expect it to evolve as the collaboration
procedes.  Promising 
technologies are described in ``Building Distributed Systems for the
Pragmatic Object Web'' 
(Fox et al., http://www.npac.syr.edu/users/shrideep/book/), and
prototypical systems have 
been investigated at JPL and NPAC.

\subsection*{Data Assimilation and Model Calibration}  Following, and in
parallel with, the code 
development GEM will build and test modules to compare observational
data with the 
models.  First will be tests for consistency between models, then data
assimilation 
capabilities will be added so that the models can be calibrated and
initialized using real data.  
Data assimilation techniques have been developed by the climate and
ocean modeling 
community that start with a given model code,  from which the ``adjoint
model'' is 
calculated.  The adjoint can be run backwards in time, beginning with
current data, to 
predict the original initial and boundary conditions that led to the
current conditions.  This 
computational procedure allows a subset of model variables to be
adjusted so that both 
initial conditions, historical observations, and current data can all be
optimally matched.  
Field data of interest include GPS, InSAR and broadband seismic
(TERRASCOPE) data, 
together with archived and newly developed paleoseismic information in
the SCEC 
database.   

\subsection*{Validation via Laboratory Experiments}  While the
collection of significant new  
laboratory data is outside the scope of the GEM project, data from
laboratory experiments 
are needed to validate the GEM simulations. Validation can take many
forms, but  there 
would seem to be two primary functions: 1) The use of  experimental data
in verifying 
numerical techniques and the physics that are included (excluded) in
models.  2) The use of 
numerical models in connection with a concurrent active program of lab
experiments to 
further identify the physical mechanisms that operate in laboratory
experiments and 
earthquake fault zones.  However, it must be recalled that laboratory
experiments provide 
information only about a strongly limited set of space-time scales on
given materials and 
conditions.  How these results scale up or apply to field situations is
unknown.   

\section*{Project Organization}

	Project organization will include the following functions:

	Planning and Coordination
	Modeling and Analysis
	Computations
	Validation/Data Assimilation
	Outreach and Information Dissemination

	It would be best if close coordination can be maintained between
ongoing projects 
and their various participants.  The optimal means of establishing such
coordination is 
probably by means of management structures associated with the Southern
California 
Earthquake Center, and its successor, the California Earthquake Research
Center.  

\section*{Priorities}

	Priorities will be determined by:

	1)  Intellectual maturity of the problem

	2)  Interests of potential funding agencies and programs.

\section*{Budget}

	While no detailed cost breakdowns have been formulated, it is possible
to envision 
an extremely basic, ``bare bones'' GEM project funded at the level of
\$100,000/year.  
Funding for a self-contained GEM project would begin at the level of
\$1,000,000/year, a 
level that would allow an active GEM Team to be formed, and to
substantively address a 
number of the tasks described within a period of about 5 years.  To make
truly significant 
progress on all of the major tasks with 3 to 5 years would probably
require a yearly level of 
funding in excess of 2,000,000/year.  

	
\section*{Figures}

The figures can be included as Encapsuled Postscript Files.

\begin{figure}[htb!]
\centerline{\epsfxsize 40 mm \epsffile{APEC_Space-time_diagram.ps}}
\caption{\protect\small Space-Time Scales of Physical Processes (from
Minster, 1998)}
\label{alabel}
\end{figure}

\section*{Acknowledgments}
\noindent
{\small
	Research upon which this paper is based has been funded by the 
US Department of Energy, the National Aeronautics and Space
Administration,
and the NationalScience Foundation.

}

\end{document}

