Scaling of Earthquake Models with Inhomogeneous Stress Dissipation

0.58 MB·English

by  Rachele Dominguez

Scaling of Earthquake Models with Inhomogeneous Stress Dissipation
∗
Rachele Dominguez
Department of Physics and Astronomy, Western Kentucky University, Bowling Green, Kentucky 42101, USA
Kristy Tiampo
Department of Earth Sciences, University of Western Ontario, London, Ontario, N6A 5B7 Canada
C. A. Serino
Department of Physics, Boston University, Boston, Massachusetts 02215, USA
1
1 W. Klein
0
Department of Physics and Center for Computational Science,
2
Boston University, Boston, Massachusetts 02215, USA

n
a Natural earthquake fault systems are highly non-homogeneous. The inhomogeneities occur be-
J cause the earth is made of a variety of materials which hold and dissipate stress differently. In

8 this work, we study scaling in earthquake fault models which are variations of the Olami-Feder-
1 Christensen (OFC) and Rundle-Jackson-Brown (RJB) models. We use the scaling to explore
the effect of spatial inhomogeneities due to damage and inhomogeneous stress dissipation in the

] earthquake-fault-likesystemswhenthestresstransferrangeislong,butnotnecessarilylongerthan
h
thelengthscaleassociatedwiththeinhomogeneitiesofthesystem. Wefindthatthescalingdepends
p
not only on the amount of damage, but also on thespatial distribution of that damage.
-
o
e PACSnumbers:
g
.
s I. INTRODUCTION mogeneities of the system [4, 5]. For long range stress
c
i transfer without inhomogeneities, as well as randomly
s distributed inhomogeneities [6] such models have been
y Thespatialarrangementoffaultinhomogeneitiesisde-
found to produce scaling similar to Gutenberg-Richter
h pendent on the geologic history of the fault. Because
p this history is typically quite complex, the spatial dis- scaling found in real earthquakesystems [7]. It has been
[ tribution of the various inhomogeneities occurs on many shown that the scaling found in such models is due to a

1 length scales. One way that the inhomogeneous nature spinodal in the limit of long range stress transfer [8, 9].
In the earthquake lattice models we use in this work
v of fault systems manifests itself is in the spatialpatterns
4 which emerge in seismicity graphs [1, 2]. we introduce inhomogeneities in the way that stress is
8 Despite their inhomogeneous nature, real faults are dissipated. Stress is dissipated both at the lattice site of
5 failure (site dissipation) and at neighboring sites which
often modeled as spatially homogeneous systems. One
3 are damaged (damage dissipation). Spatial inhomo-
argument for this approach is that earthquake faults
.
1 have long range stress transfer [3], and if this range is geneities are incorporated by varying this stress dissipa-
0 tion throughout the system in different spatial arrange-
longer than the length scales associated with the inho-
1 ments. Wefindthatthescalingfordamagedsystemsde-
mogeneities of the system, the dynamics of the system
1 pendsnotonlyontheamountofdamage,butalsoonthe
may be unaffected by the inhomogeneities. However, it
:
v is not clear that this is the case. Consequently it is im- spatialdistributionofthatdamageaswellastherelation
i ofthespatialdamageordissipationtothestresstransfer
X portant to investigate the situation in which the stress
range. Studying the effects of various spatial arrange-
transfer range is comparable to or less than the length
r
ments of site dissipation provides insights into how to
a scales associated with the damage or stress dissipation
construct a realistic model of an earthquake fault which
inhomogeneities.
is consistent with Gutenberg-Richter scaling.
In this work, we study scaling in cellular automa-
ton models of earthquake faults. We use a variation
of a model introduced initially by Rundle, Jackson and
II. MODEL
Brown(RJB)andre-introducedindependentlybyOlami,
Feder and Christensen (OFC) to explore the effect of
spatial inhomogeneities in earthquake-fault-like systems We use a two-dimensional cellular automaton model
when stress transfer ranges are long, but not necessarily of an earthquake fault which is a variant of the RJB
longer than the length scales associated with the inho- model [10, 11] and closely resembles the OFC model [5].
We beginwithatwo-dimensionallattice,whereeachsite
is either dead (damaged) or alive (active). Each live site
i contains an internal stress variable, σ (t), which is a
i
∗Electronicaddress: [email protected] function of time. All stress variables are initially below
2
a given threshold stress, σt and greater than or equal Damage Distribution γ Variance
to a residual stress σr (both of which we assume to be random 0.2510 2.1×10−4
spatially homogeneous.) Sites transfer stress to z neigh- random cascading blocks 0.2293 6.9×10−3
bors. Neighborsaredefinedasallsiteswithinthetransfer cascading dead blocks 0.2092 8.9×10−3
range, R. Initially we randomly distribute stress to each dead blocks 0.1803 2.0×10−2
site so thatσr <σ <σt. We then increasethe stress on
i
all sites equally until one site reaches σt. At this point, TABLE I: Averages and variances of γi for the distributions
ofdead sitesshown in Fig. 1. Thetotal numberof deadsites
the site at the threshold stress fails. When a site fails,
is equal to 25% of the lattice for all distributions.
somefractionofthatsite’sstress,givenbyα (σt−σr∓η),
i
is dissipated from the system, where α is a parameter
i
that characterizes the fraction of stress dissipated from
to undamaged systems with site dissipation parameter
site i, and η is a random flatly distributed noise. The α′ = 1 − φ(1 − α). These systems approach the spin-
stress of the site is lowered to σr ±η and the remaining odal (∆h→0) as the stress dissipation from the system
stress is distributed equally to the site’s z neighbors. vanishes: φ → 1 and α → 0. Physically, stress dissipa-
To model more realistic faults, we use systems which
tion from the lattice system suppresses large avalanche
are damaged, meaning they have both alive sites, which
events.
obey the rules outlined above, and dead sites which do
not hold any stress. Following Serino, et al [12], in ad-
dition to the stress dissipation regulated by the site dis-
III. QUALITATIVE BEHAVIOR OF SCALING
sipation parameter, α , we specify that any stress which
i
is passed to a neighboring dead site also gets dissipated
Firstwestudythecasewithconstantα =αanddam-
from the system. We can therefore regulate the spatial i
agedistributedinhomogeneouslythroughoutthesystem.
distributionofstressdissipationfromthesystemwiththe
In Fig. 1 we show two dimensional lattices of linear size
distribution ofthe α and the placement ofdeadsites on
i L=256and25%ofthesitesdead. Thelatticeshavevar-
thelattice. Aftertheinitialsitefailure,allliveneighbors
ious distributions of the dead sites. Figure 1(a) has the
are then checked to see if their stress has risen above σt.
dead sites randomly distributed throughout the system.
Ifithas,thissitegoesthroughthesamefailureprocedure
In the long range limit, this corresponds to homogenous
outlined above until all sites have stress below σt. The
damage studied in Ref. [12]. Figures 1(b)- 1(d) incorpo-
size of the avalanche is the number of failures that stem
ratesomeclusteringofdeadsites. Figure1(b)hasblocks
from the single initiating site. We refer to this whole
ofrandomlydistributeddeadsiteswithcascadinglength
avalanche process as a plate update.
scales where the linear block sizes range from 1 to L/8.
Because stress is dissipated from the system both at
Eachblockhasafractionpofrandomlydistributeddead
the site of failure (as regulated by α ) and through dead
i
sites, where p varies from block to block. The values
sites which may be placed inhomogeneously throughout
of p are selected from a random Gaussian distribution.
the system, we may think of each site i as having a
Figure 1(c) has dead blocks with cascading length scales
parameter which incorporates both types of dissipation,
where the linear block sizes also range from 1 to L/8.
γ =1−φ (1−α ), whereφ is the fractionofliveneigh-
i i i i Figure 1(d) has randomly distributed dead blocks with
borsofsitei. Themeanvalueγ =P γ /N ,whereN is
i i a a blocks of linear size of L/16 only. To characterize each
the number of live sites, is the average fraction of excess
configuration in Fig. 1, we calculate γ, and the variance
stress dissipated from the system per failed site.
of γ for an interaction range R=16 and α =0∀i. The
We will want to compare the scaling in these systems i i
results are summarized in Table I.
with the scaling in systems where the damage distribu-
Figure 2 shows n(s), the numerical distribution of
tion is uniform. It has been found [3] for these OFC
avalanche events of size s, corresponding to the various
type models with no spatial inhomogeneities (homoge-
distributions of damage in Fig. 1. We find that the scal-
neous damage and constant α ) that in the mean field
i
ingbehaviorofsystemswithdamagedependsnotonlyon
limit the number of avalanche events of size s obeys the
thetotalamountofdamagetothesystembutalsoonthe
scaling form
spatialdistributionofdamage. Inparticular,largeevents
n(s)∼e−∆hs/s−τ. (1) are suppressed more for lattices with damaged sites dis-
tributedmorehomogeneously. Becausethese latticesare
The quantity ∆h, which is a function of the fraction of ofequalsize,havethesamenumberofdamagedsitesand
deadsites,isameasureofthedistancefromthespinodal the same stress transfer range(R=16) the differences in
and τ =3/2. (Note that n(s) is the number of events of the large event behavior are not due to the finite size of
size s, which is the non-cumulative distribution, rather the lattice or the finite number of active sites in the lat-
than the number of events of size s or smaller, which is tice. Furthermore, the results of Sec. IIIB indicate that
the cumulativedistributionoftendiscussedinrelationto the effect is due to the spatial distribution of γ s and
i
theGutenberg-Richterlaw.) WeknowfromRef.[12]that does not even require that the lattice be damaged. The
longrangedamagedsystemswithafractionφoflivesites calculatedquantities inTable I wouldappearto indicate
andconstantsitedissipationparameterα areequivalent that the large event suppression is correlated with both
i
3
(a)random (b)random cascadingblocks
FIG.3: Numericaldistributionofavalancheeventsofsizesfor
blocksof deadsitesoflinearsize b. (Figure1(d)corresponds
to R/b = 1.) The size of the system is L = 256 and the
interaction range is R = 16. The line is drawn to show that
thedata is approaching a power law with exponent −3/2.
(c)cascading deadblocks (d)deadblocks
A. Length Scales
FIG. 1: Various configurations of 25% dead sites (in black)
For any given distribution of damage, the system will
for a lattice with linear size L = 256. Lattices contain (a)
act as if the damage is homogeneous if the stress trans-
dead sites distributed randomly, (b) blocks of various sizes,
where each block has p randomly distributed dead sites with ferrangeis longenoughcomparedtothe lengthscalesof
p varyingfor each block, (c) dead blocks of various sizes, (d) damageofthelattice. Toillustratetheimportanceofrel-
dead blocks of a single size. ativelengthscales,weconsiderthecaseofasinglelength
scale associatedwith damagedareas. We place blocks of
damaged sites of size, b, randomly in the system which
has constant α = α. See, for example, Fig. 1(d). As we
i
vary the ratio R/b, the measured value of γ varies from
γ = α for R/b ≪ 1 to γ = 1−φ(1−α) for R/b ≫ 1.
In the former case, the live domains of the system ap-
pear nearly homogeneous with φ = 1 except near the
i
boundaries of dead blocks. The latter case is the limit
of homogeneously distributed damage. In both limiting
cases,thevarianceofγ issmallandthescalingisequiva-
i
′
lenttothescalingforanundamagedsystemwithα =γ.
In Fig. 3, we compare systems with randomly dis-
tributed dead blocks of various length scales, R = 16,
α = 0, and 25% total damage. As R/b gets small, the
values of γ also get small. The distribution n of the
s
correspondingdata approachesa power law with the ex-
ponent −3/2, which is the form of the distribution of a
FIG.2: Numericaldistributionofavalancheeventsofsizesfor system at the spinodal.
various spatial distributions of dead sites. Data corresponds
to lattices in Fig. 1 with interaction range R=16.
B. Spatial Distributions of Dissipation
The spatial distribution of damaged sites determines
higher values of the averagedissipationparameterγ and thespatialdistributionofγivalues. Amoredirectwayto
lower values of the variance of γ. controlthe numericalandspatialdistributions ofγi isto
useundamagedsystemsandvarythevaluesofα . Inthis
i
In order to better understand these results, we now way,we canisolatethe effects ofspatialredistributionof
study the effect of the interaction range relative to the γi values while holding the numerical distributions of γi
lengthscalesofinhomogeneitiesandthe effectofcluster- constant.
ing of dead sites. We present data in Fig. 4 for three systems with site
4
dissipation only; that is, they have no damage and γ =
i
α for each system. We see that γ = 0.5 for all three
i
systems from the numerical distributions of α values,
i
p(α ).
i
The twosystems labeled“GaussianSplit” and “Gaus-
sian Centered” both have a uniform spatial distribution
of α values. However, as shown in Fig. 4, the values of
i
α for the “Gaussian Centered” system have a Gaussian
i
distribution centered about α =0.5, while the values of
i
α forthe“GaussianSplit”systemhavepartialGaussian
i
distributions and are clustered near the values of α =0
i
and α = 1. Thus, the variance of α values for the
i i
“GaussianCentered”systemis lessthanthe variancefor
the “GaussianSplit” system. We see fromthe numerical
distribution of avalanche events, n(s), in Fig. 4 that the
“Gaussian Split” system has slightly larger events than
the“GaussianCentered”system,consistentwiththeob-
servations above that large event suppression correlates
with low variances of α .
i
However, we see by studying the “Clustered Blocks”
system that spatial distributions of α have a robust ef-
i
fect on scaling, even when the variances of α are the
i
same. The“GaussianSplit”and“ClusteredBlocks”sys-
tems have nearly the same numerical distributions of α
i
(Fig. 4), and therefore have the same value of the vari-
ance of α . The spatial distributions of these two cases,
i
however,aredifferent: the“GaussianSplit”systemhasa
uniformspatialdistributionofα values,whilethe“Clus-
i
tered Blocks” system has high (and low) α values clus-
i
teredtogetherintoblocksasshowninthe insetofFig.4.
FIG.4: Comparison of threelattice systems with nodamage
Despite having equal values of α and equal variances
i
and two different distributions of p(αi) shown in the top fig-
of α values, the “Clustered Blocks” system experiences
i ure. Thelatticeslabeledby“GaussianCentered”and“Gaus-
much larger events (by an order of magnitude).
sian Split” are distributed uniformly in space, while the spa-
Evidently, the larger events depend crucially on the
tial distribution of “Clustered Blocks” is shown in the in-
spatialclusteringoflowdissipationsites. Thisisbecause set. The bottom plot shows the numerical distribution of
failing sites with low values of γi pass along a high per- avalanche eventsof size s.
centageofexcessstress,encouragingthe failureofneigh-
boringsites. Thus,alargeearthquakeeventismorelikely
to occur if the initial site of failure is well connected to and b is the so-called b value of the Gutenberg-Richter
a large number of sites with low dissipation parameters. law which has been measured for many real earthquake
Inoursystem,connectednessisdeterminedbyspatiallo- systems. The seismic moment M is proportional to the
cality,sowe requirelargeclumps ofsiteswith lowvalues size of the earthquake in this model [13]. Therefore, the
ofγ inordertoallowfortheoccasionallargeearthquake relation appropriate for the systems considered in this
i
event. work is the cumulative distribution of earthquake size:
N ∼s−β, (3)
s
IV. GUTENBERG-RICHTER SCALING
or the corresponding non-cumulative distribution
The Gutenberg-Richter scaling law states that the cu- ns ∼s−τ˜, with τ˜=β+1. (4)
mulativedistributionofearthquakesizesisexponentialin
themagnitude[7]. Intermsoftheseismicmoment,which Serino et al [6] construct a model for an earthquake
has succeeded the Richter magnitude as the appropriate faultsystemconsisting ofanaggregateoflattice models,
measure for earthquake sizes, the law may be reframed whereeachlattice hasafractionq ofhomogeneouslydis-
to state that the cumulative distribution of earthquake tributed dead sites and q variesfrom 0 to 1. The weight-
sizes, NM, is a power law in the seismic moment, M [6]. ing factor Dq gives the fraction of lattices with damage
q. Considering the weighting factor to be constant with
2b all values of q contributing equally to the fault system,
NM ∼M−β, with β ≡ , (2) they find a value of τ˜ = 2. They also consider a power
3
5
lawdistributionofD andfittheexponenttocorrespond affect scaling behavior even when the numerical distri-
q
to Gutenburg-Richter b values found in real earthquake butions of dissipation parameters are the same.
systems.
Therearetwoimportantdifferencesbetweenthemodel
considered by Serino et al and our work: In the model
treated by Serino et al
1. The damage is distributed homogeneously.
2. The individual lattices with homogeneous damage
q are non-interacting.
Weinvestigateboththeeffectofthespatialarrangement
ofthedamageanditsrelationtothestresstransferrange
aswellastheeffectofstresstransferbetweenregionswith
different levels of damage. (a)
We construct two lattice systems with a uniform nu-
merical distribution of γ which have scaling consistent
i
with Serino et. al.’s systems with constant distribution
D . The first model essentially pieces together many ho-
q
mogeneous lattice systems: The numerical distribution
of α values is uniform between 0 and 1, but spatially
i
arranged into N blocks of linear size B (see Fig. 5(a)
B
inset), where each block contains a random distribution
ofα valueswithinanintervalofsize1/N .Thereareno
i B
dead sites, so that α =γ . The effects of the boundaries
i i
between the blocks should be negligible if B ≪ R. In
Fig. 5(a), we present data from a system with L = 512,
R=16, and B =64. The straightline shows the best fit (b)
to a power law with exponent τ˜≃ 2.07, which is consis-
tent with the results for the aggregate lattice system of
Serino et al with Dq =1. FIG. 5: Numerical distribution for avalanche events of size s
We find that the size of the blocks, B, need not be for systems with uniform numerical distributions for αi, but
the same for different values of αi. It is important that non-uniform spatial distributions of αi which are shown in
the boxes with lower values of γ be large enough to ac- the insets. Slopes of best fit lines in red are a) τ˜≃2.07 and
commodatelargeavalancheevents,butblockswithlarge b) τ˜≃2.04.
α may be small because they are more likely to seed
i
small avalanches. With this in mind we construct a lat-
tice system with cascading length scales of blocks where Siteswithlowerstressdissipation,evenifonlypartially
the largestblocks have the lowestα values and decreas- distributedthroughoutthelatticebutclumpedtogether,
i
ing sizedblocks haveincreasingvalues of α . The scaling allowfor largeravalancheevents. We havefound models
i
results are shown in Fig. 5(b), with a best fit power law for earthquakefault systems which have avalancheevent
with exponent τ˜≃2.04. size scaling which is consistent with the new paradigm
for Gutenberg-Richter scaling proposed by Serino et al.
Themodelsstudiedheregobeyondthosepreviouslypro-
V. CONCLUSIONS posed by incorporating inhomogeneities into the lattice
and allowing areas with different characteristic dissipa-
We have studied both damage and site dissipation to tion rates to interact.
informthedevelopmentofmodelsofrealisticearthquake
faults with inhomogeneous stress dissipation. Spatially
rearranging dead sites on a given lattice affects the nu-
Acknowledgments
merical distributions of the effective stress dissipation
parameters and the scaling behavior of large avalanche
events, depending on the homogeneity of the damage This work was funded by the DOE through grant
and the length scales associated with the clustered dead DE-FG02-95ER14498 and the NSERC and Aon Ben-
sites. However, by studying site dissipation we find that field/ICLR Industrial Research Chair in Earthquake
spatial distributions of dissipation parameters crucially Hazard Assessment.
6
[1] K. F. Tiampo, J. B. Rundle, S. McGinnis, S. J. Gross, [8] J.B.RundleandW.Klein,J.Stat.Phys.72,405(1993).
and W. Klein, Europhys.Lett. 60, 481 (2002). [9] W. Klein, M. Anghel, C. D. Ferguson, J. B. Rundle,
[2] K. F. Tiampo, J. B. Rundle, W. Klein, J. Holliday, andJ.S.S.Martins,inGeocomplexity andthe physicsof
J.S.S.Martins, andC.D.Ferguson,Phys.Rev.E(Sta- earthquakes, AGU Monograph 120 (American Geophysi-
tistical, Nonlinear, and Soft Matter Physics) 75, 066107 cal Union,Washington D.C., 2000).
(2007). [10] J.B.RundleandD.D.Jackson,Bull.Seismol.Soc.Am.
[3] W. Klein, H. Gould, N. Gulbahce, J. B. Rundle, and 67 (1977).
K. Tiampo, Phys. Rev. E (Statistical, Nonlinear, and [11] J. B. Rundle and S. R. Brown, J. Stat. Phys. 65, 403
Soft Matter Physics) 75, 031114 (2007). (1991).
[4] R.Burridge and L. Knopoff,Bull. Seismol. Soc. Am. 57 [12] C.A.Serino,W.Klein, andJ.B.Rundle,Phys.Rev.E
(1967). 81, 016105 (2010).
[5] Z.Olami,H.J.S.Feder, andK.Christensen,Phys.Rev. [13] J.B.Rundle,D.L.Turcotte, andW.Klein,Geocomplex-
Lett.68, 1244 (1992). ity and the physics of earthquakes, AGUMonograph 120
[6] C. A. Serino, K. F. Tiampo, and W. Klein, 1012.1260 (American Geophysical Union,Washington D.C., 2000).
(2010).
[7] B. Gutenbergand F. Richter,Ann.Geophys. 9 (1956).