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Large earthquake production sensitivity to downdip limit of seismogenic zone determination

using subduction zone 2D computer modeling

Austin Butler

3/9/16

Introduction

The Tonga Trench, located roughly 2,000 km north-northeast of New Zealand, is a

convergent plate boundary where the Pacific plate is subducting beneath the Tonga microplate.

At its highest rate along the trench, the Pacific plate is subducting at a rate of 24 cm/yr, making it

the fastest subduction on the planet (Wright, Bloomer, MacLeod, Taylor, & Goodlife, 2000).

With such fast convergence rates, the Tonga Trench is a region of high seismicity, making it an

important and interesting region of study for the field of seismology.

Subduction zones are the cause of many of the largest earthquakes that we see. Some

convergent plate boundaries, such as the one between the Nazca and South American Plates in

western South America, occur at a continent-ocean boundary and cause earthquakes that produce

direct devastating effects. In some cases, convergent plate boundaries, such as the one in South

America, cause large earthquakes below the seafloor that then produce extremely destructive

tsunamis when a large amounts of displaced water collide with a landmass. In such cases, much

damage is done as a result of these megathrust earthquakes that greatly impacts the lives of many

around the globe. Because of this, the study of the generation of these earthquakes is very

valuable to the general population in addition to geoscientists.

A subduction zone fault model constructed by Colella, Dieterich, Richards-Dinger, and

Rubin (2012) suggests that the type of seismic activity in a subduction zone can be attributed to

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its depth on the fault plane (Figure 1). Colella et al. uses the Cascadia subduction zone to

distinguish between three zones at varying depths: the seismogenic zone, the region where the

most and largest seismic events occur, lying roughly between 5 and 25 km deep, the transition

zone lying between 25 and 40 km deep, and the continuous creep zone at depths greater than 40

km. Other similar models of megathrust faults by Liu (2013) and Lay, Kanamori, Ammon,

Koper, Hutko, Ye, Yue, & Rushing (2012) have been made that also differentiate between

different regions (Figures 2 & 3). Despite Tonga’s impressive convergence rates and frequent

seismic activity, it fails to produce great earthquakes comparable to those seen in South America,

as much of the stress is released aseismically (Beavan et al., 2010). One explanation for this is

that the downdip limit of Tonga’s seismogenic zone may not be deep enough to support

earthquakes of that magnitude. The goal of this project will be to determine if the downdip limit

of the seismogenic zone is an important control in the occurrence of large earthquakes (> 8 Mw).

To do this, I will create a model of the subduction zone megathrust based on Mohr-Coulomb

theory assuming a brittle elastic medium to look at failure propagation along the fault plane due

to changes in stress.

Tectonic Influences and Details

The driving tectonic forces that cause the subduction of tectonic plates come from

occurrences such as the formation of new oceanic lithosphere at divergent plate boundaries and

flow of the upper mantle. These forces drive plates together resulting in one of the plates at a

boundary to subduct, or to move past and beneath the other plate. The slow tectonic forces put

constant stress on the plates that are interacting. The interface between the two plates at a

convergent plate boundary has a finite strength, or resistance to these stresses. Once the stress

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has reached a critical level in a location on the interface, that location between the plates is no

longer able to support the stress, and slip occurs as the subducting plate slides past the adjacent

plate as an earthquake. This initial location of slip between plates is called the nucleation of the

earthquake. Depending on the conditions of the fault plane surrounding the nucleation, the slip

may continue, and the fault may rupture over great distances resulting in a large earthquake. The

other possibility is that a critical level of stress in the locations adjacent to the nucleation is not

reached and the rupture terminates, which is seen as a small earthquake.

The model I will produce will represent both the geometry and the physics of a

subduction zone with inspiration from the Tonga Trench. Since the seismogenic zone is limited

by depth, the dip angle of the fault would directly affect the size of the seismogenic zone, with

steeper downdip angles resulting in a smaller area of the seismogenic zone, and shallower

downdip angles yielding a larger seismogenic zone. This depth limitation of the seismogenic

zone in subduction zones is generally thermally controlled. The colder lithosphere is able to

deform brittlely, while the deeper, warmer asthenosphere supports ductile deformation. While

ruptures may propagate into the asthenosphere, this transition is generally the lower boundary for

the nucleation of earthquakes, and therefore the downdip limit of the seismogenic zone.

The dimensions of the seismogenic zone on the fault interface have an effect on the

rupture propagation of large earthquakes. After the nucleation of an earthquake, if the conditions

allow, the rupture may propagate in all directions away from the epicenter maintaining a roughly

1:1 aspect ratio. When the ruptures reach the boundaries of the seismogenic zone or regions

where brittle deformation is not supported, such as the deeper transition zone or the boundaries

of the fault, propagation cannot continue and terminates. In the case of megathrusts, the aspect

ratio of the seismogenic zone is much higher with an along-strike length much longer than the

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along-dip length. This allows for the rupture to continue propagating along-strike even though

rupture has stopped vertically. In such scenarios, larger earthquakes with magnitudes greater

than 8.5 are capable of being produced as the area of the rupture can increase horizontally.

Laboratory tests have been done to study the nucleation of earthquakes and what

characterizes the setting where they occur. Reches (1999) experimented with the triaxial loading

of both intact rock samples as well as the interaction of multiple intact samples to study both

rupture and friction, respectively, in the initiation of earthquakes. After considering instances of

both slip on pre-existing faults and rupture of intact rocks, it was concluded that it is sufficient to

assume that the rock is intact, and that the strength of the rock at different locations is must be

considered. This is in part due to the fact that under the heat and pressure conditions of the

locations of earthquake nucleation, fault zones may heal due to the re-cementation of fault

gouge. This information can be applied to my model in that it may not be necessary to

distinguish the modeled fault interface as either a pre-existing fault or completely intact rock, but

rather assume that it is intact rock with variable strengths, as this allows for the possibility that

either scenario may be the case. Another important finding was that it is not the nucleation and

growth of a single rupture that causes an earthquake, but rather the interaction of multiple

ruptures in close proximity of each other. The P wave of the main shock often followed two

smaller events, indicating that the first two triggered the main one (Figure 4). This phenomenon

can be described as the “cascade model” (Reches, 1999). In order to take this into account, my

model may require the stress threshold to be exceeded by multiple locations within a

predetermined maximum distance from each other so that a slip may cascade and result in a large

earthquake. On the same topic, a study conducted by Ohnaka (2003) had different intentions.

Ohnaka tries to solve the problem of scale-dependence of the rupture of earthquakes while taking

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into consideration the fact that rupture during an earthquake consists of both slip failure on pre-

existing faults as well as the fracture of intact rock. The result is a scaling law that “…enables

one to provide a consistent, unified comprehension for small-scale frictional slip failure and

shear fracture in the laboratory, and large-scale earthquake rupture in the field” (Ohnaka, 2003).

The idea that asperities on the subducting plate can create “locked patches” and effect the

distribution of stress on a megathrust fault interface has been explored. Konca et al. (2008)

studied the sequence of Sumatra-Andaman megathrust earthquakes in 2007 using GPS geodetic

data along the fault to analyze strain accumulation and determine the relationship between the

locked patches where coupling is high and the magnitude 8.4 and 7.9 earthquakes in 2007

(Figure 5). The inspiration for this study came from the observations that places where coupling

is low, only moderate earthquakes occur, and that places where there is high coupling, much

larger earthquakes are produced. The idea is that subduction zones where asperities lock the

plates together are able to accumulate more stress between seismic events and release more at

once than if friction between the plates was low where stress could be released more frequently

and in smaller quantities.

The use of Byerlee’s Law may prove to be helpful in setting a yield stress that will

determine when locations on the fault interface slip. Looking at friction between rocks, Byerlee

experimentally tested the shear stress required for slip to occur for a given normal stress on many

different rock types (Byerlee, 1978). His findings led to two very useful equations describing

just this scenario. In each case, the shear stress necessary for slip to occur increases linearly with

normal stress, regardless of rock type. For normal stresses less than 2 kbar, this relationship is

described as 𝜏 = 0.85𝜎!, and for normal stresses greater than 2 kbar, described as 𝜏 = 0.5+

0.6𝜎!, where τ is shear stress and σn is normal stress (Byerlee, 1978). Since the shallow end of

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the seismogenic zone lies approximately 5 km deep, it is sufficient to assume lithostatic stresses

in this environment (Stein & Wysession, 2003). At 5 km depth, the lithostatic pressure is

roughly 4.5 kbar, and the second of Byerlee’s equations can be used to determine the shear stress

where failure occurs. Byerlee notes, however, that if fault gouge is present between the two

surfaces, friction is much lower than if there were no gouge. On the other hand, asperities being

subducted, such as seamounts, can cause very high levels of friction between the plates.

Hypothesis

I predict that having a greater seismogenic zone downdip limit will enhance the

probability of large earthquakes (> 8 Mw) by allowing failure around locked patches. I will be

testing this hypothesis by systematically controlling the downdip limit of the seismogenic zone

in my 2D computer model of the megathrust fault interface. Upon analysis of my results, a

statistically significant increase in pre-defined cascading events would confirm my hypothesis,

where a statistically insignificant increase or a decrease in cascading events would falsify it.

Methods and Data

To solve the physics of the effects of stress in a brittle elastic medium, I will be

implementing the finite differences method for the differential equations of stress interaction,

taking a numerical approach. The model will consist of a rectangular grid of elements, each

representing a small area on the fault plane. Elements will experience different levels normal

and shear stresses as the “plate subducts”, and failure will occur at an element once shear stress

has overcome its force of friction. Failure and displacement of an element will then directly

affect the stresses of elements adjacent to it, which may or may not induce failure in those. A

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large earthquake would be shown as slip cascading in which failure spreads rapidly due to a

chain reaction until the strength of an element is enough to stop slip from propagating further.

All modeling will be done using Matlab.

It is important to determine the relationship between stress and strain at points (nodes) in

the continuous elastic medium I will be modeling. Assuming both isotropy and homogeneity at

the fault interface, the constitutive equation I will use for a given node is a 2-dimensional

variation of Hooke’s law,

𝜎!" = 𝜆𝑒!!𝛿!" + 2𝜇𝑒!",

where σij is the ijth component of the stress tensor; eij is the ijth component of the strain tensor; ekk

is the dilatation or the trace of the strain tensor; δij is the Kronecker delta or the ijth component of

the identity matrix; and λ and µ are the Lamé constants, the latter of which is the shear modulus.

In terms of bulk modulus, Κ, the same equation takes the form 𝜎!" = 𝛫𝑒!!𝛿!" + 2𝜇 𝑒!" −!!!!!"!

(Stein & Wysession, 2003). According to this equation, the stress and strain can be calculated

and recorded for each element of the interface. Each run of the simulation will begin with some

amount of stress being applied to a group of elements. Each element may have different material

properties and therefore may deform at different times and in different directions. A yield stress

will be calculated, and when the stress at a node exceeds this value, “slip” will occur at that

node, resulting in a dramatic and sudden decrease in stress as well as displacement of that node.

It is this displacement of the node that will quickly apply new stresses to the neighboring nodes

and will result in a higher probability that the slip will cascade and continue to propagate.

One control that I will be varying is the strength of the subducting plate, where I will be

implementing a Monte Carlo method that randomly varies strength numerous times to yield a

sufficient amount of numerical data. The strength of the plate and the interface will be

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characterized by the elastic moduli at each node, K and µ, as well as a coefficient of friction.

Also referred to as incompressibility and rigidity as well as bulk modulus and shear modulus,

respectively, K and µ will determine the deformability of the plate at each node. Elements that

are more incompressible and rigid will result in higher values of stress from the same amount of

strain experienced by elements that are more compressible and less rigid. Inversely, the same

amount of stress applied to an element with high values of K and µ will be strained much less

than elements with lower values of K and µ. Upon deformation due to stresses, the stress will be

checked after each iteration to see if it has exceeded the yield stress and if it has overcome the

coefficient of friction. The coefficient of friction at each node is a free parameter and may be

used to simulate areas of locking or areas of low friction. High coefficients of friction can

represent asperities on the subducting plate such as seamounts, rugged topography, or anything

else that may resist slip in a location. Low coefficients of friction could represent regions that

are clay-rich or have any other slip-promoting characteristics. As the relationship between stress

and strain is linear in an isotropic and homogeneous medium, Κ and µ are the constants at each

node that will convert the strain tensor to the stress tensor. By utilizing a Mohr’s circle, the yield

stress can be calculated for each node. When stress at that node exceeds the yield stress, slip

would occur and put new stress and strain on its neighboring nodes whose stress must be

checked as well, and the process repeats itself. If upon slip at a location the strength of the

neighboring locations is increased or decreased, it will be described as rate-weakening or rate-

strengthening, respectively (Figure 2). Rate-weakening, as demonstrated in the seismogenic

zone, depends upon the speed of slip, and results in a weaker fault interface as slip speed

increases (Colella, Dieterich, Richards-Dinger, & Rubin, 2012).

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A deeper downdip limit of the seismogenic zone could affect the production of large

earthquakes in several ways. First, a seismogenic zone with a longer along-dip dimension has a

larger area than a seismogenic zone with a shorter along-dip dimension, assuming equal along-

strike length. A larger area means more locations that can slip and more chances for nucleation

of earthquakes. Megathrust faults such as those in Japan, South America, and Tonga have more

than enough length horizontally, or along-strike, to produce earthquakes of enormous

magnitudes, and therefore is not likely the limiting factor in large earthquake production in

subduction zones. This is why I will be less concerned about the length of the horizontal, or

along-strike, dimension of the seismogenic zone. The aspect ratio that I will choose for my

model will be large enough that the along-strike length of the fault will not limit the potential

size of the an earthquake that can be produced. The limiting factor in the potential size of a

megathrust earthquake, rather, is determined by the strength of the fault plane that the rupture is

propagating through, which leads to another reason large earthquake production is dependent on

the along-dip dimension of the seismogenic zone. If the horizontally-propagating rupture has

more pathways in which it can circumvent high-strength locations in the fault plane, it has a

higher chance of passing around one of these strong locations and continuing to propagate

further.

The moment magnitude of an earthquake is directly dependent on the log of the seismic

moment of an earthquake. The seismic moment (M0) with dimensions of energy in dyn-cm can

be calculated using the equation

𝑀! = 𝜇𝐴𝐷,

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where µ equals the shear modulus or rigidity of the rock, a physical property of the material with

a unit of pressure; A is the rupture area along the fault plane; and D is the displacement of one of

the plates relative to the other (Stein & Wysession, 2003). Using seismic moment, the

dimensionless moment magnitude of an earthquake is calculated by the equation

𝑀! =23 log!" 𝑀! − 10.73

(Stein & Wysession, 2003). With the horizontal dimension of seismogenic zones of large

subduction zones on the order of hundreds of kilometers long and the vertical dimension only

tens of kilometers, it is clear that in order for a massive earthquake to take place, the rupture

must take advantage of the along-strike length of the seismogenic zone. Avoiding locations that

would stop the propagation of a rupture is crucial for this to happen, and is more likely to occur

with deeper seismogenic zones.

Once the simulation is run many times and ample data have been obtained, the statistical

significance of the results must be determined in order for the hypothesis to be either confirmed,

falsified, or found inconclusive. To do this, I will determine the normalized covariance between

seismogenic zone depth (independent variable) and production of large cascading events

(dependent variable) using Pearson’s correlation coefficient, 𝑟 = !!!!

(!!!!)!!

(!!!!)!!

!!!! , where xi

and yi are the seismogenic zone depth and the quantity or frequency of large events produced for

each data point, respectively; n is the number of data points; 𝑥 and 𝑦 are the algebraic means of

the independent and dependent variables, respectively; and sx and sy are the standard deviation of

the independent and dependent variables, respectively. Once the covariance is determined, its

significance can be determined by using a resampling method such as jackknifing or

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bootstrapping. This will provide an objective result to the data and will aid in the conclusion of

the project.

Discussion

Upon completion of my project, the results must be analyzed in order to come to a

conclusion on my hypothesis. As this is a computer-modeling project, many iterations may be

done in a small amount of time to test many different scenarios and different variables. Contrary

to other projects relying on physical experiments, changes to the model can be made without

entirely starting the model-building process over. With this being said, trial and error can lead to

the production of many different models and will allow them to eventually evolve into a

physically very accurate one. The processes being replicated in the model show an example of

deterministic chaos, similar to the results of the slider block experiment (Turcotte, 1997). As

some computer models may come to the same result given the same initial conditions,

deterministic chaos implies that a result cannot be predicted, even if the initial conditions are

known, although the result may be known within limits (Figure 6). This requires that the model

must be run many times and the results of all the iterations looked at at once. This may allow

determination of the probability of an event occurring given certain initial conditions.

In order to test my hypothesis, I must model subduction zones with a variety of aspect

ratios and areas. This will be testing the effect of the downdip limit or the along-dip length of

the seismogenic zone on generating great earthquakes. The number or frequency of cascading

events and large earthquakes for each scenario must be determined and compared. In order for

my hypothesis to be validated, there must be a significantly larger amount of cascade events in

model runs corresponding to deeper seismogenic downdip limits than in runs corresponding to

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shallower downdip limits. If my hypothesis is proven to be correct, this could potentially explain

the relative absence of great earthquakes seen on the Tonga megathrust despite its high

convergence rates and frequency of seismicity. If no significant relationship is seen between

seismogenic zone downdip limit and large earthquakes, it could rule out downdip limit as a

controlling factor in the production of large earthquakes, and a different explanation must be

proposed to explain Tonga’s lack of great earthquakes.

Timeline

The following is an approximation of what will need to be finished at what time in order

to stay on track to graduate at the beginning of the Summer 2016 semester:

March

• Construct 2D model of subduction zone

• Begin writing of thesis

April

• Finish model and thesis

• Prepare presentation for thesis defense

Late April/Early May

• Defend thesis

May

• Make any changes to thesis and/or model for reevaluation if necessary

• Submit corrected thesis to graduate school

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Figures

Figure 1 - A megathrust fault model based of off the Cascadia subduction zone differentiating between 3 distinct regions of slip (Colella et al., 2012)

Figure 2 - A cross-section of a subduction zone showing locations of velocity- (or rate) weakening and velocity-strengthening slip (Liu, 2013)

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A

B

Figure 3 – A.) A to-scale cross-section of the subduction zone off the eastern coast of Japan differentiating between 4 distinct regions of slip B.) A more detailed look at the slip on the fault interface of the 4 regions shown in 3A. (Lay et al., 2012)

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A

B

Figure 4 – A.) Velocity seismogram schematic where the P-wave of the main shock followed a separate initial phase of multiple smaller events (left), and the slip history of the 3 events (right) B.) A series of steps showing the cooperation of nearby cracks leading to the growth of a fault in intact rock (Reches, 1999)

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Figure 5 – Locations of strong coupling (red regions) on the Sunda megathrust due to subducted asperities creating locked patches correspond to locations of large earthquakes (Konca et al., 2008)

Figure 6 – Results of an asymmetrical two-block slider block model showing chaotic behavior (Turcotte, 1997)

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Works Cited

Beavan, J., Wang, X., Holden, C., Wilson, K., Power, W., Prasetya, G., … Kautoke, R. (2010).

Near-simultaneous great earthquakes at Tongan megathrust and outer rise in September

2009. Nature, 466(7309), 959–963. http://doi.org/10.1038/nature09292

Byerlee, J. (1978). Friction of rocks. Pure and Applied Geophysics PAGEOPH, 116(4-5), 615–

626. http://doi.org/10.1007/BF00876528

Colella, H. V., Dieterich, J. H., Richards-Dinger, K., & Rubin, A. M. (2012). Complex

characteristics of slow slip events in subduction zones reproduced in multi-cycle

simulations. Geophysical Research Letters, 39(20), n/a–n/a.

http://doi.org/10.1029/2012GL053276

Konca, A. O., Avouac, J.-P., Sladen, A., Meltzner, A. J., Sieh, K., Fang, P., … Helmberger, D.

V. (2008). Partial rupture of a locked patch of the Sumatra megathrust during the 2007

earthquake sequence. Nature, 456(7222), 631–635. http://doi.org/10.1038/nature07572

Lay, T., Kanamori, H., Ammon, C. J., Koper, K. D., Hutko, A. R., Ye, L., … Rushing, T. M.

(2012). Depth-varying rupture properties of subduction zone megathrust faults. Journal of

Geophysical Research: Solid Earth, 117(4), 1–21. http://doi.org/10.1029/2011JB009133

Liu, Y. (2013). Numerical simulations on megathrust rupture stabilized under strong dilatancy

strengthening in slow slip region. Geophysical Research Letters, 40(7), 1311–1316.

http://doi.org/10.1002/grl.50298

Ohnaka, M. (2003). A constitutive scaling law and a unified comprehension for frictional slip

failure, shear fracture of intact rock, and earthquake rupture. Journal of Geophysical

Research, 108(B2), 1–21. http://doi.org/10.1029/2000JB000123

Reches, Z. (1999). Mechanisms of slip nucleation during earthquakes. Earth and Planetary

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Science Letters, 170(4), 475–486. http://doi.org/10.1016/S0012-821X(99)00122-3

Stein, S., & Wysession, M. (2003). An Introduction to Seismology, Earthquakes, and Earth

Structure (1st ed.). Blackwell Publishing.

Turcotte, D. (1997). Fractals and Chaos in Geology and Geophysics (2nd editio). Cambridge

University Press.

Wright, D. J., Bloomer, S. H., MacLeod, C. J., Taylor, B., & Goodlife, a. M. (2000). Bathymetry

of the Tonga Trench and Forearc: A map series. Marine Geophysical Researches, 21(5),

489–511. http://doi.org/10.1023/A:1026514914220