Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
Models of stochastic, spatially varying stress in the
crust compatible with focal mechanism data
Deborah Elaine Smith
Thomas H. Heaton
Corresponding Author:
Deborah Elaine Smith
1120 Seneca Place
Diamond Bar, CA 91765
1
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
2
Abstract
Evidence suggests that slip in earthquakes and their resultant stress changes are
spatially heterogeneous. If crustal stress from past earthquakes is spatially heterogeneous
then earthquake focal mechanisms should also be spatially variable. We describe the
statistical attributes of simulated earthquake catalogs (including hypocenters and focal
mechanisms) for a spatially 3-dimensional, time-varying model of the crustal stress
tensor with stochastic spatial variations. It is assumed that temporal variations in stress
are spatially smooth and are primarily caused by plate tectonics. Spatial variations in
stress are assumed to be the result of past earthquakes, and are independent of time for
periods between major earthquakes. It is further assumed that heterogeneous stress can
be modeled as a stochastic process that is specified by an autocorrelation function.
Synthetic catalogs of earthquake hypocenters and their associated focal mechanisms are
produced by identifying the locations and times at which the second deviatoric stress
invariant exceeds some limit. The model produces a seismicity catalog that is spatially
biased; the only points in the grid that exceed the failure stress are those where the
heterogeneous stress is approximately aligned with the stress rate. This bias results in a
focal mechanism catalog that appears less heterogeneous than the underlying stress
orientations. Comparison of synthetic focal mechanism catalogs with catalogs of real
earthquakes suggests that stress in the crust is strongly heterogeneous. Stochastic
parameters are estimated that generate distance dependent spatial variations in focal
mechanisms similar to those reported by Hardebeck [2006] for Southern California.
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
3
Introduction
Statistical Paradigm for Stress Heterogeneity
There is a great deal of complexity in seismically active regions in the Earth that
can lead to spatially heterogeneous slip and stress. Specifically, the crust in these regions
is often highly fractured by faults of varying lengths. On the fault themselves, there can
be repeated slip events ranging from 10 meters to less than several millimeters. Given
this highly complex spatial and temporal slipping history, it follows that the stress
distribution should also be highly complex or spatially heterogeneous.
One question that naturally arises is how can we characterize this spatially
heterogeneous stress in the crust? Given the complex slip history of the Earth’s crust at
widely varying scales, it is not reasonable to deterministically track stress changes from
every dynamic event over the entire spatial bandwidth. Instead, as a first approximation,
we assume that the spatially varying stress heterogeneity is statistically stationary for the
inter-seismic period (between major events) and can be described with fractal-like
statistics using two parameters. This represents the stress pattern left over from the
combined action of previous earthquakes. We also assume that during the inter-seismic
period, stress from tectonic forces grows steadily in time and smoothly in space where
this steady stress change is superimposed on the rough fabric of stress heterogeneity.
Another question that arises from considering spatially heterogeneous stress, is
what effect does stress heterogeneity have on stress inversion estimates of background
stress (the spatial mean stress within a region)? When one uses earthquakes within a
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
4
selected region, with the associated focal mechanism parameters, and applies a stress
inversion to estimate background stress, one assumes that the included events are a
uniform sampling of the initial stress field. That works well if the initial stress field is
approximately homogenous spatially within the selected region; however, does the
assumption of uniform sampling work well in a spatially heterogeneous stress field given
that earthquakes are critical phenomena? We show that no, in a spatially heterogeneous
stress field, earthquakes are a biased sampler of the initial stress field (mean stress +
stress heterogeneity), producing biased stress inversion results.
Hence, our goal in this paper is to develop stochastic models of 3D deviatoric
stress heterogeneity that will: 1) produce focal mechanisms statistics that match those
seen in the real Earth and 2) demonstrate a bias toward the tectonic stressing rate in stress
inversion results that is a function of the stress heterogeneity.
We develop models with only two statistical parameters,  , which is related to
the fractal dimension or spatial roughness of stress, and HR (Heterogeneity Ratio),
which describes the ratio of spatial heterogeneity to the spatial mean. We then generate
synthetic earthquakes and their associated focal mechanisms from these numerical
models, by applying an appropriate failure criterion. We will show that the different
values of  and HR have distinct affects on the synthetic seismicity catalogs. By
comparing the orientation and clustering statistics of our synthetic data sets with a catalog
of focal mechanisms in Southern California, we are able to produce an estimate of our
statistical stress parameters  and HR . We then use these synthetic data sets, with a
focus on our preferred  and HR , to estimate the orientation biasing in stress inversion
results for Southern California.
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
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Our model to create heterogeneous focal mechanism orientations contrasts
strongly with many previous studies that assume stress is approximately homogeneous
and failure occurs only on randomly oriented planes of weakness (pre-existing faults). In
particular, numerous studies of the statistical properties of focal mechanisms have
inferred the orientation of an approximately spatially uniform stress that best aligns with
the possible slip vectors that are compatible with focal mechanism catalogs [Carey and
Brunier, 1974; Angelier, 1975; Etchecopar, et al., 1981; Angelier, 1984; Gephart and
Forsyth, 1984; Michael, 1984; 1987; Mercier and Carey-Gailhardis, 1989; Gephart,
1990]. While these homogeneous stress models can apparently explain many features of
focal mechanism characteristics, we will demonstrate that it is also possible to explain
these statistics with our model, which is quite different; i.e. in our model the stress is
heterogeneous, the temporally varying stress (the stress rate tensor, which is due to plate
tectonics) is relatively homogeneous, and the frictional strength is assumed to be uniform.
It seems clear that these two classes of models (homogeneous stress and
heterogeneous strength vs. heterogeneous stress and homogeneous strength) represent
end members, and that the real Earth likely has both heterogeneous stress and
heterogeneous strength [Rivera and Kanamori, 2002]. While each of these models allow
researchers to infer Earth processes from focal mechanism catalogs, we will show that the
model of heterogeneous stress and homogeneous strength provides very different
information than can be obtained from the traditional stress inversion studies.
Evidence for Spatially Heterogeneous Stress
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
6
Observations of spatially varying slip along fault zones and in earthquakes
suggest that both slip and stress are very spatially heterogeneous and possibly fractal in
nature [Andrews, 1980; 1981; Herrero and Bernard, 1994; Manighetti, et al., 2001; Mai
and Beroza, 2002; Ben-Zion and Sammis, 2003; Lavallee and Archuleta, 2003;
Manighetti, et al., 2005]. For example, McGill and Rubin [1999] observed a one meter
change in slip over distances of approximately one kilometer in the Landers earthquake,
which indicates a 103 strain change. This implies possibly a 100 MPa stress change
averaged over the distance of 1 kilometer. Similar strain changes can be seen in the slip
inversion of seismic and geodetic data from the Landers earthquake [Wald and Heaton,
1994] .
Another example of highly variable, heterogeneous slip over short wavelengths
comes from Manighetti et al. [2001]. Using altimetry data in the Afar depression, East
African rift, they show heterogeneous cumulative slip as a function of distance, with
short wavelength strains of the order 5x102 . While non-elastic processes may come into
play at such large shear strains, these observations of heterogeneous slip demonstrate a
few features. Heterogeneous slip patterns exist not just for individual earthquake slip
histories but persist for the entire cumulative slip history of fault zones, indicating that
slip heterogeneity is a stable feature. In addition, the cumulative slip shows possibly selfsimilar, fractal patterns [Manighetti, et al., 2001; Manighetti, et al., 2005].
Borehole studies, which measure the orientation of maximum horizontal
compressive stress, S H , directly from borehole breakouts, also indicate that stress can be
quite heterogeneous. Wilde and Stock [1997] reported on multiple boreholes with
different orientations that had been drilled at approximately the same locations. Wilde
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
7
and Stock analyzed this data to constrain the relative magnitudes of the principal stresses.
Boreholes drilled within close proximity of each other (less than one kilometer) show
greatly varying S H orientations (tens of degrees), which may be indicative of
heterogeneous stress (See Figure 1). The orientations of borehole breakouts in the Cajon
pass borehole [Barton and Zoback, 1994] also shows significant heterogeneity for an
individual borehole near an active fault. The orientation data are reproduced later in this
paper (Figure 14) together with a comparison of our stochastic stress.
Liu-Zeng et al. [2005] have also shown that the assumption of short wavelength
heterogeneous fractal slip can reproduce distributions of earthquakes having slip vs.
length ratios similar to real earthquakes and realistic Gutenberg-Richter frequency
magnitude statistics. Using simple stochastic models, they showed that spatially
connected slip can produce averaged stress drops (a constant times average slip divided
by rupture length) similar to real data.
Perhaps the most interesting piece of data comes from Zoback and Beroza [1993];
they studied the orientations of aftershock planes from the Loma Prieta earthquake and
plotted their distributions as a function of strike and dip. They found aftershocks that had
both right-lateral and left-lateral orientations on similar fault planes as well as normal and
reverse orientations. Given that this seismicity is in the immediate vicinity of the rightlateral San Andreas Fault, the existence of left-lateral aftershocks on fault planes parallel
to the San Andreas Fault presents a curious problem. Zoback and Beroza proposed that
the principal compressive stress direction was almost normal to the San Andreas Fault
and that the aftershocks occurred on extremely weak faults of different orientations
surrounding the mainshock zone. However, if one allows for a new paradigm of spatially
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
8
heterogeneous stress in three dimensions, as presented in this paper, the left-lateral
orientations naturally occur.
Furthermore, the model we present predicts that such opposite mechanisms
should primarily be observed for aftershocks. Figure 13 shows our hypothesis for what a
1D cross section of shear stress in Southern California might look like. While most of
the points have positive shear stress on the  12 plane, a fair percentage have negative
shear stress on the  12 plane. Heterogeneity similar to this could explain why Zoback
and Beroza observed left-lateral aftershocks after the Loma Prieta earthquake; the large
local stress change to the system from the mainshock, combined with stress heterogeneity
in the left-lateral direction, would create the left-lateral aftershocks.
Motivation for Heterogeneous Stress
Assuming that fractal-like spatially heterogeneous slip and even more
heterogeneous stress occur in the Earth and that this heterogeneity is compatible with
seismic observables, one may ask, “How does the Earth produce this spatial
heterogeneity?” Numerical modeling has revealed at least a few potential sources for
creating stress heterogeneity in the Earth from dynamic earthquake models that include
varying friction [Aagaard and Heaton, 2008] to kinematic models of slip on
geometrically complex faults [Dieterich, 2005; Dieterich and Smith, 2009, in press].
Aagaard and Heaton [2008], produced slip heterogeneity for some of their
simulated dynamic earthquake ruptures. In particular, they found that slip heterogeneity
increased as the dynamic ruptures became more pulse-like, where pulse-like behavior
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
9
tends to be produced when there is a strong slip velocity dependence in the friction law
[Aagaard and Heaton, 2008].
A strong slip velocity dependence in the friction law, upon which pulse-like
behavior and dynamically produced slip/stress heterogeneity depends, is supported by
both theoretical and experimental work. Rice [1999; 2006] for example, suggests that
sudden transitions in dynamic friction may be plausible in the presence of flash heating
on the slip surfaces. Tullis and Goldsby [2005], observe dramatic reductions in sliding
friction for velocities  50 cm s and suggest flash heating as a possible mechanism.
Beeler, Tullis, and Goldsby [2008] show plots of friction coefficient as a function of
sliding velocity for three different materials: quartz, granite, and gabbro, and again see
significant reductions in the sliding friction. While the low velocity friction coefficients
range from a little over 0.6 to approximately 0.9 depending on the material, at sliding
velocities of  50 cm s , the sliding coefficient of friction approaches a value of 0.2.
These experiments also observe instantaneous full healing when the slip velocity
decreases. This combination of high static friction, low sliding friction, and
instantaneous healing back to high static friction may freeze in short length-scale
dynamic stress variations.
A static effect that can also produce heterogeneous stress has been presented by
Dieterich [2005; Dieterich and Smith, 2009, in press]. Fault traces in nature are rarely if
ever completely planar; there is usually some small-scale 3D geometry to the fault trace.
Modeling fault traces with fractal geometry, solving for slip with boundary elements and
using a   0.6 , they find that even very small variations from planarity in fault trace can
produce significant near-fault stress heterogeneity and create spatially heterogeneous
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
10
aftershock rates. In this case, the coefficient of friction was not varied dynamically, so
this is an entirely independent source of stress heterogeneity.
Stress Model to Be Used
We develop the following fractal-like model of crustal stresses in space and time.
We assume that the fractal-like statistics are statistically stationary; hence, this
description is intended only for interseismic times, the periods of time between large
earthquake sequences. In our numerical models of stress, we construct a threedimensional spatial grid in which the stress is determined at every grid point as follows,
 



 x,t   B  &T t  t0   H x
(1)
where
 B is the background stress, which is the spatially and temporally averaged stress
tensor in the region of interest. This term,  B , minus any overall stress
magnitude information, is the intended solution of stress inversions; i.e. stress
inversions attempt to solve for the three principal orientations and the stress ratio,
   3 
R 2
 [Rivera and Kanamori, 2002], of  B .
 1   3 
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton

11

&T t  t0 is the temporally varying stress due to plate tectonics that brings points
within our 3D grid to failure as point earthquakes. t0 is the time of the last large
earthquake; t0 is arbitrarily chosen to be the start time of any of our simulations.
We assume that temporal variations in stress are primarily caused by forces that
are applied at a distance and that the temporally varying stress is approximately
spatially homogeneous. In principle, this temporally varying stress can be
observed directly from geodetic data and knowledge of the elastic properties.
While it is clear that temporal variations in stress do change with location, the
observed spatial variations are small compared with spatial heterogeneities that
arise from heterogeneous slip in past earthquakes.
This term is assumed to grow linearly with time for our short simulation

time windows of 10–20 years, but is assumed to be small compared to  H x
and  B . In general, we assume that  B and &T may have different orientations.
For example, the principal compression of the average background stress may be
oriented nearly perpendicular to the San Andreas Fault [Townend and Zoback,
2004]; whereas, the stress rate compression axis must be at a 45° angle, since
shear on the San Andreas Fault accommodates most of the plate motion.

 H x is spatially varying stress. By definition, its expected value is zero. The
heterogeneous stress is assumed to be due to all of the stress changes caused by
local inelastic deformations, such as the slip distribution due to faulting,
compaction, fluids, thermal stresses, topography, and other factors. We assume
that the heterogeneity is described by two parameters,
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
12
1.  , where the amplitude spectrum of any 1D cross section through our 3D

 H x grid is proportional to 1
kr
where kr is wave-number [Barnsely,
et al., 1988],
2. Heterogeneity Ratio, HR , is a measure of the relative amplitudes of the
heterogeneous stress compared with the uniform background stress. We
measure the amplitude of the stress tensor using the scalar inner product of
the deviatoric stress tensor with itself. This inner product (denoted by : ) is
equivalent to the second scalar invariant of the deviatoric stress tensor,
I 2   ij :  ij , where  ij   ij   ij kk 3 denotes the deviatoric stress
tensor. I 2 is a measure of the shear strain energy density.
HR 
I 2 H
I 2 B



  H x :  H x 


 B :  B
Where I 2 H is spatial average of the second invariant of the heterogeneous
deviatoric stress. Since I 2 H is the sum of the squared components of the
deviatoric stress tensor, I 2 H is the sum of the variances of the components

of  H x .
This particular parameterization of the relative size of the background
stress (a constant) and the heterogeneous stress (a spatially varying fractallike term) depends on the number of points used in our grid. Therefore, it
is unavoidable that any parameterization of the relative size of the
(2)
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
13
background stress and the heterogeneous stress depends on the number of
points in our finite grid. [One way around this, is to specify the
resolution at which you sample the stress grid and be
consistent…(think about this)]
Assuming the fractal-like statistics are statistically stationary for inter-seismic

times, means the stochastic properties of  H x , described by HR and  , do not
significantly evolve in time for the simulations presented in the paper; therefore,

we do not update  H x after each event. Our focus is to compare our results to
stress inversions applied to background seismicity in-between major seismic
events over a time window in the range of 1–20 years. The significant 3D
heterogeneous stress changes that a major earthquake event introduces would
have to be taken into account, to study the aftershock period.
Assumptions/Limitations of This Stress Formulation
From the outset it is important to indicate clearly the assumptions used in this
paper and the possible limitations. We do not attempt to create stress heterogeneity in 3D
from first principals because of the inherent difficulties. Aagaard and Heaton [2008]
present numerical simulations of self-sustaining heterogeneous slip and stress on a 2D
plane. Their simulations were severely limited in spatial bandwidth by the immense
numerical calculations that are required to faithfully create realistic 3D stress
heterogeneity. Furthermore, derivation of stress derived from dynamic modeling requires
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
14
many assumptions, such as the distributions of fault orientations, fault lengths, dynamic
friction on faults, etc. Instead, we have chosen to approach this problem with a simple
statistical model. On the plus side, this enables us to describe spatially heterogeneous
stress with two statistical parameters, HR and  , generate synthetic focal mechanisms
quickly, and compare our simulations with real data to constrain the statistical properties
of the crust. On the other hand, this statistical approach makes many simplifying
assumptions in an attempt to obtain insight into the nature of stress variations in the
Earth’s crust and overlooks details that are necessary if one wishes to model stress
heterogeneity from first principles.
In particular, our spatial model of stress does not satisfy the equations of static


equilibrium. While we satisfy rotational equilibrium  ij   ji , we do not satisfy the
other static equilibrium equation,  ij, j  f i , where f i is a body force. In order to satisfy
 ij, j  f i , the solutions to which are spatially smooth, and to have spatially
heterogeneous stress, we would have to include sources, which requires a whole set of
additional assumptions. Of course, it is the local effect of sources that is the likely origin
of heterogeneous stress, but we are trying to avoid the complications of explicitly
including these sources.
The other assumptions for the formulation presented in this paper include: 1) there
is no such thing as pre-existing planar faults in our model, which means that our
seismicity tends to cluster in 3D clouds rather than lineations or planes, as seen in the real
Earth.
2) When failure occurs at a grid point, the stress only changes at that grid point
(it actually drops out of the simulation once it has failed). This means that there is no
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
15
explicit interaction between events. This assumption is clearly inappropriate for large
events that change stress over a large region. However our stress model given by
equation (1) is intended to simulate background seismicity, where stress perturbations
due to individual events are small and should have little to no effect on the other events
included in the regional inversions. 3) We assume failure occurs on fresh-fracture,
maximally oriented planes at 45 from the  1 and  3 principal stress axes. This is a
consequence of using a plastic yield criterion. In Appendix C from Smith [2006], we do
use a Coulomb Failure criterion and find similar but more complicated results when we
compare our results for Coulomb Failure criterion to our results for the plastic yield
criterion. 4) Lastly, our stress heterogeneity is fractal-like; its outer scale is set by the
box size and the inner scale is set by the resolution, rather than scales that might exist in
the Earth.
Numerical Simulations of Fractal Stress
The purpose of this paper is to explore the consequences of spatially
heterogeneous stress on the orientations of focal mechanisms. In order to do this, we
specify the stress tensor everywhere in our 3-dimensional grid. Furthermore, we assume
that this stress tensor varies stochastically in space, and that the spatial variation is
statistically stationary and thus it is described by a correlation function (or equivalently
by a power spectrum). We further require that the principal orientations of the
heterogeneous stress tensors are statistically isotropic; that is, we do not want to introduce
preferred orientations by an arbitrary choice of our coordinate frame. Six independent
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
16
quantities are required to describe the stress at each grid point, and these six quantities
vary spatially according to a correlation function in such a way that the grid is
statistically isotropic. Although there may be many ways to achieve this, we investigated
two alternative representations of the six independent quantities that have the desired
properties.

Three tensor eigenvalues (the principal stresses,  H 1 ,  H 2 ,  H 3 , which are
scalar invariants of the stress) and three angles that describe the orientation of the
principal coordinate frame. In Smith [2006] the three angles chosen were  ,  ,
 , similar to Euler pole rotations where one rotates by angle  about an axis
specified on a sphere by angles  and  . This representation has the advantage
that statistical properties are defined for scalar invariants. Unfortunately, this
representation is somewhat cumbersome to use.

Six independent Cartesian stress tensor components  H 11 ,  H 22 ,  H 33 ,  H 12 ,
 H 23 ,  H 13 . While this representation is relatively simple to use, the Cartesian
components are not invariant with respect to the coordinate frame.
Enforcing isotropy requires special attention in both approaches, and we found that
assuming a Gaussian distribution of principal stresses produces Cartesian components
that are described by a beta distribution. Likewise, assuming a statistically isotropic
stress whose components are described by a Gaussian distribution produces principal
stresses that are described by a beta distribution. For a complete discussion of results
using the eigenvalue/eigenvector filtering, see Smith [2006]. It is unclear which
methodology better captures the actual physics in the Earth, and the estimated stochastic
parameters are similar between the two methodologies, so for this paper, it is sufficient to
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
17
present the results for the simpler of the two approaches, the Cartesian component
analysis.
For each of the 6 independent components of the Cartesian stress tensor, we
assigned values to each grid point using a Gaussian random number generator. The
variances of the diagonal terms were assumed to be one and the variances of the offdiagonal terms were assumed to be 1
2
; this scaling is required to produce isotropy
[Albert Tarantola, written communication]. We then applied a spatial filter to produce
the desired spatial correlation function. Because the heterogeneous stress grid has no
preferred coordinate frame, we can uniquely specify its correlation properties by defining
the spectral properties along any line that bisects the grid. If kr is the wave–number
along any line in which r  xi xi specifies distance along the line, then we filter the grid
such that the expected value of the amplitude spectrum along any line is
E ̂ H ij kr   1  kr
E ̂ H ij kr  


1
1  kr
2
i j


i j
This form leaves the zero-frequency properties unchanged, while filtering the short
wavelength variations according to a power law. This produces a fractal-like distribution
that is relatively free of inherent length scales. See Figure 2 for 1-D examples of filtered
stress and their associated amplitude spectrums. In Figure 3, we show the effect of
filtering on the stress orientations. For a 1-D line, we plot all the possible three angles,
(3)
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
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 ,  ,  , associated with a stress tensor relative to a reference stress tensor where  is
represented by color,  is the co-latitude and  is the longitude.
In order to achieve the spectrum given by equation (3) for our 3-D grid, we
Fourier transformed our spatial grid of random numbers with respect to the three


Cartesian coordinates to obtain φij k1 , k2 , k3 . We then filtered with a 3-dimensional


wave–number filter Fφ k1 , k2 , k3 specified by

F̂ k1, k2 , k3   1  ki ki


 f  ,n 3
(4)

f  , n  3 is a function that is required to produce 1-D spectra with a power law decay
of kr  from a grid of n dimensions (in our case, n  3 ). By trial and error we found that
 
f  , n can be approximated by
   1.87  2 
f  ,n  2  1.53 
 1  1,
 1.57 

(5)
   3.5  2 
f  ,3  2.97 
 1  1,
 3.33 

(6)


and
 
Although our discretized grids are fractal-like, they have an inner scale (the grid spacing)
and an outer scale (the length of the grid), and thus they are not true fractals. In Figure 4,
we apply equations (3), (4), and (6) to the components of our stress tensors with initially

random Gaussian distributions in our 3-D grid. We then plot one component,  H 11 x ,
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
19
along a 2-D cross-section of our 3-D grid, to show how the filtering equations effect
stress on a plane.
The Average Stress Tensor

Our heterogeneous stress,  H x , has been constructed such that the expected
value of every component is zero, which means that there are no preferred orientations of
the stress. However, this does not mean that the spatially averaged stress tensor is the
null tensor. Consider the case of uncorrelated white noise with variance of unity (   0 ).
Then the expected value of the amplitude of the mean of N points is 1
N
. Although
the expected amplitude of the mean is small for large N when   0 , it becomes
increasing larger as  increases; the distribution becomes dominated by long wavelength
components and the mean amplitude increases.

Now we have designed our model of stress,   x , such that we specify the

background stress,  B , with the intention that it represents the spatial mean of   x . In

order to accomplish this, we rotate  H x so that the principal axes of  H are aligned
with the principal axes of  B . We then add a constant stress  C such that
 B   C   H .
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
20
Fracture Criterion Used
To create synthetic failures and their associated earthquake focal mechanisms, we
need to select an appropriate fracture criterion. Our preferred fracture criterion is the
Hencky-Mises plastic yield condition [Housner and Vreeland, 1965] because of its
simplicity. It predicts failure when I 2 , which is proportional to the shear strain energy, a
scalar quantity that is related to the maximum shear stress, is greater than a threshold
value. Since I 2 and the maximum shear stress are invariant quantities, this failure
criterion works regardless of the coordinate system or orientation of the individual stress
tensors. The coefficient of friction is essentially zero (optimally oriented planes), and
pressure does not enter into the equation. (If one wishes to investigate non-zero pressures
and coefficients of friction see Appendix C, Coulomb Fracture Criterion, in Smith
[2006].) Last, because we are dealing with optimally oriented planes, the conjugate
planes become mathematically indistinguishable.
Failure occurs when
2
I 2  x, t    02
3
(7)
[Housner and Vreeland, 1965], where  0 is the uniaxial yield stress and I 2  x, t  is the
second invariant of the deviatoric stress tensor,   x,t  , where
I 2  x, t     x, t  :   x, t  .
(8)
We show in another paper [Smith and Heaton, in preparation 2008] how the interaction
between a fracture criterion and spatially heterogeneous stress results in hypocenters that
coincide with locations where the local stress is aligned with the stress rate tensor, T ;
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
21
that is, this model produces a biased sampling of all of the stress tensors in the grid. The
objective of this paper is to compare the statistical properties of synthetic focal
mechanisms generated by our spatially heterogeneous stress with statistical properties of
focal mechanism catalogs compiled for real earthquakes. Hopefully, this may shed some
light on the characteristics of stress heterogeneity in the real Earth.
Given the fracture criterion in equation (7), definition of I 2  x, t  in equation (8),
and equation for deviatoric stress tensor (1), we show in Figures 6 and 7, the effect of our
two statistical parameters, HR and  , on generating failures and their associated focal
mechanisms. In Figure 5, we represent plots of the Von Mises stress,  VM 
3
I 2 , as a
2
function of position along lines that pass through our grid and for different values of 
and Heterogeneity Ratio, HR . We also plot a horizontal dashed line that represents a
hypothetical failure threshold for our Hencky Mises plastic failure criterion. As time
increases in our model, the tectonic stress rate tensor causes changes in stress at every
grid point, the Von Mises stress increases at some points and decreases at others
depending on the alignment of the orientations of   x,t  and T . When the Von Mises
stress exceeds the yield threshold then an event is declared for that grid point.
An important feature of this model is that the synthetic seismicity catalog is a
biased sample of our 3D grid; the points in the grid that have principal stresses aligned
with the stress rate tensor, T , will experience the largest temporally increasing shear
strain energy which can eventually lead to failure. This can be seen in Figure 6 where we
compare the distribution of 20,000 compression axes that are randomly sampled within
our grid in the top row with the distribution of 20,000 compression axes that are produced
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
22
by synthetic earthquakes (a biased distribution) in the middle and bottom rows. In all
cases, we assume that there is a spatial correlation of   0.7  . In the middle row, we
have let the stress rate tensor,  align with the spatial mean background stress,   and
in the bottom row,  is rotated 45° with respect to   . The three distributions shown
are a) HR  0.1 , which corresponds to weak spatial heterogeneity HR  100 , b) HR  1 ,
which corresponds to moderate heterogeneity, and c) HR  100 , which corresponds to a
highly heterogeneous distribution HR  0.1 . Notice that when the heterogeneity is large,
the earthquake sampled distributions in the middle and bottom rows are far from a
uniform random sample of the space. When  and   are aligned (as in the middle
row), the distribution of orientations clusters around   with less scatter than one would
expect for the given heterogeneity ratio, HR . When  is rotated relative to   , the
earthquake distributions for c) in the bottom row are heavily biased towards the


orientation of the stress rate tensor even though & t  t0 is small compared to the other
terms in equation (1).
In Figure 7, we plot map projections of synthetic focal mechanisms that occur
within our grid. The nine maps represent three different values of both heterogeneity
ratio, HR , and the spatial correlation parameter,  . In general, when there is little to no
heterogeneity (small HR ) the orientations of focal mechanisms are nearly identical (the
panels on the left). As HR increases, so does the variation of focal mechanism
orientations (the panels on the right). When  is small, there is little to no correlation in
the spatial locations and orientations of the focal mechanisms. However, as seen in the
bottom right panel for large HR and  , when  is larger, there is spatial clumping of
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
23
the focal mechanism locations, and the orientations tend to be similar for focal
mechanisms located near one another.
Model Parameter Estimates for the Earth
Can our simple statistical model of stress replicate key features of real seismicity
and can it reveal the statistical characteristics of stress heterogeneity in the Earth? To
answer this we generate suites of 3D spatially heterogeneous stress grids with different
values of the spatial correlation parameter,  , and heterogeneity ratio, HR , to produce
synthetic focal mechanism catalogs. We then compare these synthetic focal mechanism
catalog statistics to A and B quality events from a 1984–2003 Southern California data
set [Hardebeck and Shearer, 2003] (www.data.scec.org/research/altcatalogs.html). The
comparison of synthetic and real focal mechanism statistics allows us to estimate which
pair of stress heterogeneity parameters,  and HR , best fits the real data for Southern
California.
We first have to account for observational uncertainty in comparing our synthetic
focal mechanisms catalogs to the Southern California data set. The two uncertainties we
include are: 1) uncertainty in the focal mechanism orientation,  FM , and uncertainty in
the hypocentral location,  hypo . We assume that both of these errors are Gaussian with a
mean of zero that we add to our synthetic focal mechanism orientations and positions.
Although Hardebeck and Hauksson [2001] find that exponential noise best models the
mechanism uncertainty/error in focal mechanisms, we find that there is little to no
difference in our results between mean of zero Gaussian noise (normal distribution noise)
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
24
vs. exponential noise. We also find that heterogeneity in catalog parameters produced by
catalog errors trade off with intrinsic stress heterogeneity; noisy data can be mistaken as
heterogeneous stress.
We compare our synthetic data to Southern California focal mechanisms two
ways. First, we use a grid search technique over four parameters,  , HR ,  FM , and
 hypo , to find a set of  and HR within a specified uncertainty that best fits a plot of
average angular difference between focal mechanisms as a function of pair separation,
shown by Hardebeck [2006]. Her result is calculated from the A and B quality events in
the Southern California data set. Second, we use the mean misfit angle statistical
parameter,  from stress inversions of focal mechanisms to compare our synthetic
catalogs to the real Southern California data. This measure, which only constrains HR ,
produces results consistent with the previous grid search results of  and HR .
The first statistical measure we use, by comparing our synthetic results to
Hardebeck’s plot [2006], is average angular difference,
Average Angular Difference  AAD 
N
N
1
  ij ,
N  N  1 i 1 j 1 for ( j i )
where ij is the magnitude of the rotation to rotate the ith focal mechanism into the
jth orientation. See Figure 8, where her original results for the Southern California
region using the hypoDD (+3D) method is plotted as a thin black line and used as the
background. The thick lines on top represent our 3D simulation results that
(9)
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
25
approximately fit the Southern California curve. As we will later show, this statistical
measure provides constraints on the spatial correlation parameter,  . Hardebeck [2006]
observed that average angular distance, AAD, decreased with inter-event distance until
leveling off at AAD ≈ 26° for distances less than 100 m (for the Southern California
region). She reasoned that AAD should approach zero as the inter-event distance became
small, and she interpreted her observation to mean that the average uncertainty of focal
mechanism orientations is 26°. However, our model assumes that stress heterogeneity
can occur at all length scales, even those that are smaller than the location error. That is,
even if we assume no uncertainty in focal mechanism determination (i.e.,  FM  0 ) our
model will naturally produce a non-zero AAD as R  0 , simply because of uncertainties
in R .
Although there is merit in exploring the complete range of models that can be
produced by our four model parameters  HR,  ,  FM ,  hypo  , it is time consuming and
beyond the scope of this study. However, we used trial and error to attempt to find a
model that is compatible with the observations of Hardebeck and Shearer and [2003]
Hardebeck [2006]. Not surprisingly, we found that there is a tradeoff between the focal
mechanism error and the location error. Although neither error is actually known, we
fixed the location error at 50 m, and then deduced a focal mechanism error of 12o.
We find that  HR  2.5,   .70,  FM  12o ,  hypo  50m  provides an acceptable fit
to the data. Because the wide bandwidth of Hardebeck’s study (10 m to 100 km) would
require an unmanageably large 3D grid with 1012 points, we extended our 3D models
using a simple 1D grid model for shorter wavelengths. This is possible since we
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
26
constructed our statistical parameters in a way that they have the same numerical values
in 1D as they do in 3D.
Figure 9 shows the perturbations in the solution as a function of perturbations
from the preferred solution for each of our four model parameters. Although there are
model tradeoffs, each of the model parameters has a different effect on the predicted plot
of AAD as a function of reported inter-event distance. Notice that Hardebeck’s AAD
trends towards 60o for inter-event distances larger than 10 km. This can be compared
with an AAD of 75o that is expected for a random uniform distribution of planes (such as
seen in Figure 11a). In our model, AAD at large inter-event distances is a function of
both HR and  FM .
The second statistic is the mean misfit angle,  , which is often used in focal
mechanism stress inversion studies. Typically, stress inversion algorithms derive the
spatially uniform stress tensor that is most compatible with a set of focal mechanisms.
The misfit angle for an individual event in the catalog is defined as the angle between 1)
the shear traction vector that is the result of projecting the mean stress tensor onto one of
the conjugate planes and 2) the slip vector for that same plane.  is simply the mean of
the misfit angles.  can be a function of focal mechanism measurement error, fault
plane ambiguity, and actual stress heterogeneity. Hence,  is a function of HR and  FM ,
but it is not affected by either  or  hypo .
In order to compare our results with observed data, it is important to process our
catalog of events in the same manner that stress inversion studies use to derive stress
orientations and statistical measures, such as  . Therefore we use the stress inversion
program slick [Michael, 1984; 1987] to invert catalogs of synthetic focal mechanisms
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
27
produced by our model (the synthetic focal mechanisms include  FM ). Figure 11b shows
 as a function of the heterogeneity ratio, HR, and for two different values of  FM .
In stress inversion studies, it is common to assume that the average uniform stress
varies from one region to the next, presumably due to variations in the geometry of plate
motions. To use the program slick to compare our model to actual data, we sub-divided
southern California into regions and used slick to invert A and B quality mechanisms
from the Hardebeck and Shearer catalog [2003], just as if we were conducting a focal
mechanism stress inversion study of southern California. The regions are typically about
50 x 50 km except for the Ventura Basin, Los Angles Basin, and San Gabriel Mountains.
See Figure 10 for a map of the regions studied and Table 1 lists the mean misfit angle
results with a range of 21.4° to 30.8°. These values taken together with the derived mean
misfit angle  , in Figure 11b implies a HR in the range of 1.41 to 3.08, with a weighted
value of approximately 2.2. We also calculate the AAD statistic for these 12 regions, list
the results in Table 1, and derive an HR range of 1.12 to 3.50 with a weighted value of
approximately 2.1 by comparing these AAD values to the derived AAD as a function of
HR in Figure 11a. These values for HR are quite close to the HR value estimated from
Figure 8 of 2.5. We haven’t included all the data in these estimates that was included in
Figure 8, which could explain the slight discrepancy.
 Figure 12 compares P-T plots of real focal mechanisms and simulation focal
mechanisms. On the left, we plot P-T axes for 300 A and B quality focal mechanisms
using the Hardebeck and Shearer [2003] catalog in the Banning region (see Figure 10).
The right panel shows P-T axes produced by our model  FM using our parameters from
Figure 8,  HR  2.5,   .70,  FM  12o  . The synthetic P-T axes were generated using a
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
28
background stress,  B , with horizontal compression and tension axes oriented, North 20°
West/North 70° East and a stressing rate, T , with horizontal compression and tension
axes oriented, North 25° East/North 65° West. In other words, the P-T axes for T , are
rotated 45° from the P-T axes for  B , about the vertical axis. The failure mechanisms
produce P-T axes almost half-way between those associated with  B and T ,
demonstrating again the biasing effect shown in the bottom row of Figure 6. There is a
rough similarity between the real P-T axes on the left and synthetic P-T axes on the right.
The Size of Stress at Different Scales
In Figure 13 we plot a 1D cross section for one component of the deviatoric stress
tensor, using our average best guess parameters for Southern California. We create 1D
heterogeneous stress with ten million points and an   0.70 ; we then add the following
background stress tensor,
0 1 0


B   1 0 0 
0 0 0 ,


normalizing this background stress and our heterogeneity so that we have an HR  2.5 .
We equate one grid spacing to one cm; therefore, our entire spatial bandwidth is
approximately 100 km, or seven orders of magnitude. We set the maximum stress at 200
MPa, which is approximately the expected shear strength of granitic rock [Scholz, 1990]
as measured in a laboratory for a 10-cm sample.
Notice that our preferred stochastic model implies that there are regions of stress
oriented backwards from the spatial mean. Or in other words, we expect to find local
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
29
regions of left-lateral stress on right-lateral faults, or local regions of normal stress on
thrust faults. Normally, these backward stress regions would not be detected, plate
tectonics would serve to take these regions further from failure as time progresses.
However, the occurrence of a large near-source stress perturbation from a significant
earthquake, may coincidentally cause increased backwards stress in regions of preexisting backwards stress, thus triggering a small number of backwards aftershocks. This
could explain the observation of Zoback and Beroza [1993] where left-lateral aftershocks
were seen on the San Andreas Fault after the 1989 Loma Prieta earthquake. Indeed,
one’s estimate of mean shear stress can radically change depending on the length-scale
over which we average as well as the location.
If the stress in the Crust has any similarity to Figure 13, then how should we
define the strength of the Crust? Clearly, it doesn’t make much sense to define it as the
peak stress achieved at a local site. On the other hand, if we define strength to be the
amplitude of the average stress, then this amplitude depends on the length scale over
which the averaging occurs. This can be seen in Figure 13; the amplitude of the average
stress steadily declines from ~ 164 MPa at a length scale of 100 m when we focus on an
asperity to ~ 40 MPa at a length scale of 100 km for the entire grid. These values are
compatible with the closed form solutions of Heaton and Elbanna [2008, In preparation];
they derived the length scale dependence of the expected value of the amplitude of shear
stress for a material with a stochastic stress. If the stress is described by a Fourier
amplitude spectrum with a 1D spectral decay of 1  kr  , then the amplitude of the

average stress should scale as L1 . They also discuss the length scale dependence of
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
30
strength in chaotic systems that self-organize into heterogeneous stress configurations
that described by their autocorrelation functions (i.e. Fourier amplitude spectra).
Our last comparison is shown in Figure 14 we plot the orientations of borehole
breakouts in the Cajon Pass borehole as reported by Barton and Zoback [1994] on the
left and then in the middle and on the right we plot synthetic borehole breakout data from
a grid using our best guess parameters,   0.70 and HR  2.5 . The middle and right
panels use the same synthetic data. The difference is that in the right panel some of the
data has been thrown out and a modeled azimuthal measurement error with 5° standard
deviation has been added. In both the real data and the synthetic data, there are
significant rotations in the borehole breakouts over a relatively short length-scale of 300
m. However, the short wavelength heterogeneity is stronger in the reported data than is
predicted by our preferred model.
Discussion and Conclusions
While it may be possible to explain the divergence of focal mechanisms by
appealing to the proper assemblage of very weak faults embedded in a homogeneous
stress field, there is little laboratory evidence that some faults nucleate earthquakes at
much lower shear stress than other faults. We have presented an opposite alternative
explanation of these divergent mechanisms; that is, a relatively uniform spatial
distribution of strength (i.e. nucleation stress) together with spatially heterogeneous
stress. Furthermore, there is clear evidence that stress changes in earthquakes are
strongly heterogeneous, from which we can surmise that the stress may be heterogeneous
as well.
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
31
We have described a very simple stochastic model of heterogeneous stress in the
Crust; in essence it only has two parameters, one that describes the autocorrelation
function as a power law, and the other that describes the relative amplitude of
heterogeneous stress compared with the regional mean. It is clear that this simple
description cannot capture the diversity of failure characteristics observed in the Crust of
southern California. For example, seismicity along the San Jacinto Fault is scattered and
is described by the Gutenberg-Richter law, whereas the Carrizo Plain segment of the San
Andreas Fault is largely devoid of small earthquakes. Clearly, this model cannot
simultaneously describe both of these regions. Furthermore, the stochastic parts of our
model are statistically isotropic, an unlikely scenario when we clearly observe regions
with coherent structures such as the Transverse Ranges. Nevertheless, this simple model
introduces new physical principles that must apply if stress is spatially heterogeneous. In
particular, it is hard to avoid the conclusion that seismicity provides a biased spatial
sample of the Crust. That is, earthquakes will only occur in those locations that have
stresses that are favorable aligned with stress changes from tectonic activity (or from
stress changes caused by nearby large earthquakes). This implies that a random sample
of stress orientations in the Crust is even more divergent than is observed in catalogs of
focal mechanisms.
The implications of strongly heterogeneous stress are numerous. In a second
study [Smith and Heaton, in preparation 2008], we show that when focal mechanism
stress inversion codes are run using focal mechanism catalogs produced by our model,
then the stress inversion produces a stress orientation that is an intermediary between the
orientation of the average background stress tensor and tectonic stress rate tensor.
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
32
Our seismicity model can be modified to include aftershock orientations by
introducing a spatial pattern of stress change from a main shock within our spatial grid.
In this case Smith and Dieterich [submitted 2008]show that aftershocks are characterized
by a larger diversity of focal mechanisms and by an average orientation that can be
different from that of the background seismicity. However, this effect only persists
during the aftershock period.
Finally, it seems that we need to develop some more flexible definition the word
strength. That is, earthquakes apparently occur at many length scales in a material whose
average stress depends on the length scale. Our preferred (and admittedly simple) model
has strong heterogeneity; the size of the spatial variation is larger than the average stress.
Furthermore, we find evidence that the stress may have an amplitude spectrum that scales
as k r 0.7 . Heaton and Elbanna [2008, In preparation] shows that this implies that the
amplitude of the stress averaged over length scale, L , should approximately scale as
L0.3 .
Data and Resources
• Focal mechanism data used was A and B quality events from 1984–2003 Southern
California data set [Hardebeck and Shearer, 2003]. This catalog is documented at the web
site, www.data.scec.org/research/altcatalogs.html, “2005 - Hardebeck: focal mechanisms
from P-wave polarity and S/P amplitude ratios.
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
33
• Stress inversions of focal mechanisms were all done using Andy Michael’s program
“slick” [Michael, 1984; 1987]. This program is available for download at the following
web link, http://earthquake.usgs.gov/research/software/index.php.
• Some of the plots were generated using GMT, “Generic Mapping Tools,” available at
http://gmt.soest.hawaii.edu/.
• All other code was developed using Matlab and run on a G5 computer with OS X.
Acknowledgements
We would like to thank the Southern California Earthquake Center and the National
Science Foundation for helping fund this project. We also thank Jeanne Hardebeck for
our discussions with her and for the use of her focal mechanism data/results for Southern
California. Thank you to Joanne Stock for her detailed and insightful comments. Last,
we would like to especially thank Albert Tarantola for his fruitful advice and discussions
on how to create a spatially stochastic, fractal-like description of the 3D stress tensor.
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mechanism data – Smith and Heaton
36
Smith, D. E., and T. H. Heaton (in preparation 2008), How focal mechanism inversions,
applied to earthquakes in a stochastic, spatially varying stress field, can produce solution
stress tensors biased toward the stress rate.
Townend, J., and M. D. Zoback (2004), Regional tectonic stress near the San Andreas
fault in central and southern California, Geophysical Research Letters, 31, 1-5.
Tullis, T. E., and D. L. Goldsby (2005), paper presented at Chapman Conference on
Radiated Energy and the Physics of Earthquake Faulting, Portland, Maine, June 13-17,
2005.
Wald, D. J., and T. H. Heaton (1994), Spatial and temporal distribution of slip for the
1992 Landers, California, earthquake, Bulletin of the Seismological Society of America,
84, 668-691.
Wilde, M., and J. Stock (1997), Compression directions in southern California (from
Santa Barbara to Los Angeles Basin) obtained from borehole breakouts, Journal of
Geophysical Research, 102, 4969-4983.
Zoback, M. D., and G. C. Beroza (1993), Evidence for near-frictionless faulting in the
1989 (M 6.9) Loma Prieta, California, earthquake and its aftershocks, Geology, 21, 181185.
Author Affiliations and Addresses
Deborah Elaine Smith
Earth Sciences Department
University of California, Riverside
Riverside, CA 92521
desmith@ucr.edu
desmith144@gmail.com
(work) 951-827-6041
Thomas H. Heaton
Seismolab 252-21
California Institute of Technology
Pasadena, CA 91125
Heaton_t@caltech.edu
626-395-4232
Table Captions
Table 1. Summary of the statistics for our twelve regions in Figure 11. The average
angular difference, AAD , was calculated for each region and compared to the curve in
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
37
Figure 12 a) to generate the associated Heterogneity Ratio, HR , estimates. The mean
misfit angle,  , for each region was calculated through a type of bootstrapping, were the
points in each region were inverted fifty times using a different sampling each time, 
was calculated for each inversion, then the average  for the fifty inversions is listed in
this table. Last, by comparing these values to Figure 12 b) we create their associated
HR estimates. Using the AAD statistic, we have HR range from 1.12 to 3.5 with a
weighted mean of 2.13. Using the  statistic, we have an HR range from 1.41 to 3.08
with a weighted mean of 2.21. These weighted values are close but slightly smaller than
our preferred HR  2.5 from Figure 9.
Statistics for the Twelve Study Regions in Southern California
Ventura Basin
San Gabriel
Mountains
LA Basin
Apple Valley
Landers
Banning
Palm Springs
Coachella
Northern Elsinore
Central Elsinore
Anza
Borrego
Number
of Points
1201
62
Average Angular
Difference (Degrees)
57.6860
61.1852
HR for
FM = 12°
1.9
2.58
Mean Misfit
Angle (Degrees)
24.9866
21.3923
HR for
FM = 12°
1.87
1.41
107
324
247
1089
706
63
133
111
404
64
62.0778
46.2063
63.8005
58.6933
62.9247
56.9429
53.3378
52.9828
54.0552
54.3786
2.84
1.12
3.5
2.05
3.1
1.84
1.53
1.48
1.59
1.61
28.1707
23.3583
30.7571
29.1196
28.8701
21.4996
22.2927
24.0690
22.8804
25.2197
2.51
1.68
3.08
2.7
2.65
1.43
1.5
1.77
1.61
1.89
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
38
Table 1
Figure Captions
Figure 1. Wilde and Stock [1997] plotted inferred maximum horizontal compressive
stress, SH , orientations from borehole breakouts in Southern California. There are a
variety of orientations for borehole breakouts from the same borehole or from boreholes
spatially close to one another. This suggests short-wavelength spatial stress
heterogeneity. In this modified plot, we have used circles to point out a few of the
locations studied by Wilde and Stock that show evidence for SH orientation heterogeneity.
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
39
Figure 2. a) Gaussian white noise in the top panel for a 1D line with a spatial
correlation parameter,   0.0 (equivalent to no filtering). This could for example, be
  The bottom panel is a log-log
one component of a heterogeneous stress tensor,   x
plot of the Fourier amplitude spectra of this noise vs. its spatial frequency. b) Gaussian
white noise filtered with   0.5 in the top panel. The bottom panel is the log-log plot of
the Fourier amplitude spectrum of this filtered noise vs. its spatial frequency. Note that
the slope of the trend ≈ -0.5. This is the desired 1D slope for a spatial correlaiton
parameter of   0.5 .
Figure 3. Plots of stress tensor orientations for 1D lines, 100,001 points each with
different spatial correlation parameters,  , applied. All 16 possible rotations from a
single reference stress tensor to the 100,0001 stress tensors in the filtered line have been
plotted. The rotations are represented by the three angles,  , which is the amplitude of
the rotation and  ,   which is the axis of rotation.  ,   defines the spatial co-latitude
and longitude of the points in the plots and  defines the color. Interestingly, for
  0.0 , the orientations and colors are fairly uniformly distributed. As the spatial
correlation increases, the spatial clustering and color grouping of the points increases,
until for   1.5 , you can distinctly see lines on the unit sphere representing orientations.
This is simply showing the fairly smooth variation of stress tensor orientations along the
modeled 1D line when   1.5 .
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
40
Figure 4. Plots of a scalar tensor component,  H 11 x , with different filters applied.
The 2D cross sections are x-y planes through approximately the center of the 3D grid.
We start with Gaussian noise and apply   0.0, 0.5, 1.0 and 1.5 to produce scalar
tensor components with different spectral 1D falloffs.
Figure 5. 1D cross sections of the Von Mises stress for different values of our two
parameters,  (our spatial filtering parameter) and HR (Heterogeneity Ratio which
compares the amplitude of the spatially heterogeneous stress at some outer scale with the
amplitude of the spatial mean stress). On the left, curves for   0.0 are plotted and on
the right, curves for   1.0 are plotted. The Von Mises stress is plotted where,
 VM 
3
I 2 , and represents the distortion strain energy. The solid, dashed black line
2
on each plot represents the maximum distortion strain energy the material can withstand
before yielding and producing an event, namely an earthquake. On each plot, the Von
Mises stress for three different Heterogeneity Ratios, HR , are plotted on top of one
another. The largest ratio, HR  1.0 , is plotted in blue on the bottom, the ratio
HR  0.35 is plotted on top in red, and the smallest ratio, HR  0.1 , is plotted on the
very top in green. They are normalized so that at time, t  0.0 , only the top 5% points
exceeds the failure threshold. One can see that the larger the HR , the greater the
amplitude variation in the Von Mises stress as a function of length. Similarly, the
orientations of the total deviatoric stress tensor,   x,t  , will have greater variation as
HR increases.
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
41
Figure 6. Nine equal area plots, with 20,000 P axes each. It shows the effect of
increasing Heterogeneity Ratio, HR , for three different scenarios and how synthetic
simulations are a biased sampling of the initial 3D grid of over 8 million points. In the
first row, we uniformly sample 20,000 points in our 3D grid, apply three different values
of HR , and assume maximally oriented failure planes for the stress tensor,  B   H x  .
When the heterogeneity is small, the failures tightly cluster around the P axis associated
with  B . When the heterogeneity is large (and  H x  has little to no average
orientation), the P axes approximately uniformly cover the equal area plot. The second
and third rows are P axes for synthetic focal mechanisms, which are a biased sampling of
our 3D grid. This biasing can produce two effects, a tighter than predicted clustering of
failures and an orientation bias toward the stressing rate term,  T . In the second row,
we have let
T

 B . In this case, the P axes will always cluster around the
&T
B
orientation compatible with  B and the only biasing effect in play is the decreased P
axes scatter for large HR . In the third row, we rotate  T 45° with respect to  B about
the vertical axis. Now we have the additional biasing effect that as HR increases, the
failures produce P axes increasingly biased toward an orientation compatible with the
stressing rate,  T , even though the  T contribution to the stress sum directly is one part
in a thousand or less.
Figure 7. Plots of synthetic focal mechanisms produced by our numerical model. Note
that any orientation randomness in this figure is due purely to stress heterogeneity. We
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
42
have not yet added noise due to measurement uncertainty of focal mechanism
orientations that one sees in real data. Our two numerical model parameters,  and
HR , are varied. For small HR , such as HR  0.1 in the left panel, the orientations of
the failures are nearly identical independent of the spatial smoothing parameter,  . As
HR increases so does the randomness in the focal mechanism orientations. For small
values of  , such as   0.0 in the top panel, the spatial locations of the failures are
almost uniform random in their distribution. The orientations of the focal mechanisms
are also fairly uncorrelated spatially. As  increases, the spatial correlation of the
stress field increases affecting both the locations and orientations of the failures. The
failure locations begin to clump together in space, and the focal mechanisms that are
close to one another tend to have similar orientations.
Figure 8. Background figure for each panel is modified from Hardebeck [2006]. The
thin black line for Southern California is Hardebeck’s HypoDD (+3D) solution for this
region. The average focal mechanism difference increases with distance between focal
mechanism pairs, indicating there is some type of smoothed heterogeneity. We calculate
combined 1D and 3D simulations that seem to best fit the curves and plot our results on
top of Hardebeck’s data. The 3D solution, averaged over 50 simulations (ten different
random seed 3D grids with five different sets of noise added for each grid), is plotted
with a thick solid line and the 1D solution averaged over 50 simulations (50 different
random seed grids with one set of noise added to each), is plotted with a thick dashed
line. By trial and error, we find a reasonably good fit with
HR  2.5,  .70,
FM

 12o ,  hypo  50m .
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
43
Figure 9. These figures show the effect of our four parameters on our 3D simulation
curve in Figure 8. Specifically, we look at what happens when we vary: 1) the spatial
correlation parameter,  , 2) the heterogeneity ratio, HR , 3) the hypo central location
uncertainty,  hypo , and 4) focal mechanism orientation uncertainty,  FM .
Figure 10. These are the 12 regions we studied (where the San Gabriels are lumped
together as one region). We took A and B quality focal mechanisms from the Hardebeck
and Shearer catalog [2003], and ran two types of statistics. We applied the program
“slick” [Michael, 1984; 1987] to each region to invert for a best fitting stress tensor and
associated mean misfit angle,  . We also calculated the average angular difference,
AAD , between pairs of focal mechanisms. By comparing the statistics for data from
these 12 regions to our derived relations for AAD vs. HR and  vs. HR in Figure 11, we
produce additional estimates for HR in Southern California.
Figure 11. Curves relating a) the average angular distance, AAD , to HR and b) the
mean misfit angle statistic,  , to the heterogeneity ratio, HR . The solid black curves
were generated by calculating AAD or  for synthetic focal mechanism data sets with a
focal mechanism uncertainty of,  FM  12 , for different heterogeneity ratios, HR .
When generating panel b), the relation between  and HR , we randomly sampled the
data set for each HR 50 times for 50 separate inversions, and plotted the mean  for
each HR . The thin solid horizontal lines, represent the AAD and  values from Table
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
44
1 for the twelve regions we studied in Figure 11. Last, the grey shaded columns
represent the intersection of our data for the twelve regions and the HR curves
calculated from our synthetic data. This gives us an estimate of the HR range
compatible with our real data.
Figure 12. P and T axes for 300 events in Banning on the left. On the right, we plot 300
synthetic focal mechanisms each, using our best guess stress heterogeneity parameters,
HR  2.5,  .70,
FM

 12o . For the synthetic events on the right, we assumed a
background stress tensor,  B , with horizontal compression and tension axes oriented at
North 20° West and North 70° East. The stess rate tensor,  T , was assumed to have
horizontal compression and tension axes oriented at North 25° East and North 65° West
(i.e. rotated 45° from  B ).
Note the rough similarity between the real and synthetic
data with regards to the spread of the P-T axes.
Figure 13. Spatially smoothed heterogeneous stress 1D profiles with   0.70 and
HR  2.5 . We calculate ten million points of one component of the deviatoric stress
 . If we let the grid spacing equal 1 cm, then the entire range of our stress 1D
tensor,  12
cross section is approximately 100 km. In each plot we only show 10,000 points by either
zooming in on an asperity or by sampling the grid. In a) we plot the entire width, a 100
km length. In b), c), and d) we successively narrow our plotting window by an order of
magnitude each time, to focus in on a stress asperity. The mean shear stress increases as
we decrease the plotting window, showing that the strength of the material can be lengthscale dependent; namely, longer ruptures, which sample a large percentage of the entire
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
45
1D profile, will have a smaller mean stress drop (due to averaging effects) than smaller
ruptures if the smaller ruptures simply reflect small localized stress asperities.
Figure 14. Comparison of real and synthetic borehole breakout data. On the left is
borehole breakout data from the Cajon pass in a particularly heterogeneous section
[Barton and Zoback, 1994]. In this figure, modified from Barton and Zoback [1994],plus
signs are used to plot the maximum horizontal compressive stress, SH , azimuth as a
function of depth and triangles are used to plot their model. In the middle panel we plot
SH for a moderately noisy section of synthetic crustal stress heterogeneity with our
preferred parameters of   0.70, HR  2.5 . The panel on the right modifies the
synthetic data in the middle panel to better compare with the data of Barton and Zoback.
Specifically, we generated an independent filtered line with our preferred parameters
  0.70, HR  2.5 , and used that to spatially determine which points would be kept;
all points above zero were kept and those below zero were thrown out. Last, for the
remaining points we added a random azimuthal noise to mimic measurement error with a
standard deviation of 5°. This modified synthetic data in the right panel captures similar
features as the real data in the left panel, including rapid changes in SH of over 90° over
10s of meters.
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
46
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
Figure 1
47
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
Figure 2
48
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
  0.0
  0.7
  1.0
  1.5
Figure 3
49
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
  0.0
  0.7
  1.0
  1.5
Figure 4
50
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
Figure 5
51
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
HR  0.1
HR  1.0
Uniform
Sampling
of
 B   H x 
Synthetic
Simulations
where
&T

 B
&T
B
Synthetic
Simulations
where
&T

 B
&T
B
Figure 6
HR  100
52
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
Figure 7
53
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
Figure 8
54
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
Figure 9
55
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
Figure 10
56
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
a)
b)
Figure 11
57
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
Figure 12
58
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
Figure 13
59
Models of stochastic, spatially varying stress in the crust compatible with focal
mechanism data – Smith and Heaton
Figure 14
60